On the quantum affine vertex algebra associated with trigonometric R -matrix
aa r X i v : . [ m a t h . QA ] J a n ON THE QUANTUM AFFINE VERTEX ALGEBRA ASSOCIATEDWITH TRIGONOMETRIC R -MATRIX SLAVEN KOˇZI ´C
Abstract.
We apply the theory of φ -coordinated modules, developed by H.-S. Li, tothe Etingof–Kazhdan quantum affine vertex algebra associated with the trigonometric R -matrix of type A . We prove, for a certain associate φ of the one-dimensional additiveformal group, that any φ -coordinated module for the level c ∈ C quantum affine vertexalgebra is naturally equipped with a structure of restricted level c module for the quan-tum affine algebra in type A and vice versa. Moreover, we show that any φ -coordinatedmodule is irreducible with respect to the action of the quantum affine vertex algebraif and only if it is irreducible with respect to the corresponding action of the quantumaffine algebra. In the end, we discuss relation between the centers of the quantum affinealgebra and the quantum affine vertex algebra. Introduction
The notion of vertex algebra , originally introduced by Borcherds [2], presents a remark-able connection between mathematics and theoretical physics. The vertex algebra theoryled to important breakthroughs in multiple areas such as automorphic forms, finite sim-ple groups and soliton equations; see, e.g., the books by E. Frenkel and Ben-Zvi [15], I.Frenkel, Lepowsky and Meurman [18] and Kac [24]. Some of the most extensively studiedexamples of vertex algebras come from the theory of affine Kac–Moody Lie algebras; seethe books by Kac [23] and Lepowsky and Li [28]. Motivated by a parallel between the de-velopment of the theories of affine Lie algebras and quantum affine algebras, as well as byfurther applications to two-dimensional statistical models and the quantum Yang–Baxterequation, I. Frenkel and Jing [17] formulated a fundamental problem of generalizing thevertex algebra theory to the quantum case.The notion of quantum vertex algebra was introduced by Etingof and Kazhdan [11]based on the ideas of E. Frenkel and Reshetikhin [16]. The examples of quantum ver-tex algebras were constructed in [11] as quantizations of the quasiclassical structure onthe universal affine vertex algebra in type A when the classical R -matrix is of rational,trigonometric and elliptic type. Recently, a structure theory of quantum vertex algebraswas developed by De Sole, Gardini and Kac [5] and the Etingof–Kazhdan constructionwas generalized to the rational R -matrix in types B , C and D by Butorac, Jing andthe author [3]. On the other hand, several more general related notions, in particular, of h -adic nonlocal vertex algebra and of its module , were introduced and extensively studiedby Li [30]. They present analogues of the corresponding notions, coming from the Li non-local vertex algebra theory [29] and the Bakalov–Kac field algebra theory [1], which aredefined over the commutative ring C [[ h ]], thus being compatible with Etingof–Kazhdan’stheory. Moreover, the notion of h -adic nonlocal vertex algebra module, which presentsa generalization of vertex algebra module, appears to provide the right setting for thestudy of representations of double Yangians and of Etingof–Kazhdan’s quantum vertex Department of Mathematics, Faculty of Science, University of Zagreb, Bijeniˇckacesta 30, 10 000 Zagreb, Croatia
E-mail address : [email protected] .2010 Mathematics Subject Classification.
Key words and phrases.
Quantum affine algebra, Quantum vertex algebra, φ -Coordinated module,Quantum current. lgebras associated with the rational R -matrix; see [30] and [27] respectively. However,Li’s subsequent results [31] suggest that the solution of the original Frenkel–Jing problemof associating quantum vertex algebras to quantum affine algebras requires a new con-cept of φ -coordinated module . Following such an approach, Li, Tan and Wang [32] recentlyestablished a correspondence between restricted modules for the Ding–Iohara algebra oflevel 0 associated with the affine Lie algebra b sl [8] and φ -coordinated modules for certainquantum vertex algebra.The definition of a φ -coordinated module W for a quantum vertex algebra V , as givenin [31], is characterized by a certain deformed version of the weak associativity property.Roughly speaking, it requires that the expressions (cid:0) ( z − z ) p Y W ( u, z ) Y W ( v, z ) (cid:1)(cid:12)(cid:12) z = φ ( z ,z ) and ( φ ( z , z ) − z ) p Y W ( Y ( u, z ) v, z )coincide for all u, v ∈ V , where Y ( z ) is the vertex operator map on V , Y W ( z ) the φ -coordinated module map, φ ( z , z ) ∈ C (( z ))[[ z ]] an associate of the one-dimensionaladditive formal group and p > u, v . While setting φ ( z , z ) = z + z leads to the usual weak associativity property, a different choice of the associateappears to be required in order to adapt the theory to quantum affine algebras; see [31].Let g N = gl N , sl N . In this paper, we consider the quantum affine vertex algebra V c ( g N )associated with the trigonometric R -matrix, as defined by Etingof and Kazhdan [11]. Weshould mention that V c ( g N ) can be also regarded as an associative algebra over C [[ h ]],which is topologically generated by the coefficients of certain Taylor series organized intothe matrix T + ( u ) ∈ End C N ⊗ V c ( g N )[[ u ]], subject to certain dual Yangian-type definingrelations. As a quantum vertex algebra, its vertex operator map Y ( z ) is given in the formof quantum currents T ( u ), which go back to Reshetikhin and Semenov-Tian-Shansky[35]. Furthermore, the S -locality property for V c ( g N ), which is a quantum analogue ofthe locality in the corresponding affine vertex algebra, comes from the quantum currentcommutation relation which, in this particular setting, can be expressed as T ( u ) R ( e − u + u − hc ) T ( u ) R ( e − u + u ) − = R ( e − u + u ) − T ( u ) R ( e − u + u − hc ) T ( u ) , (0.1)where R ( x ) = R ( x ) is the trigonometric R -matrix of type A . On the other hand, theoriginal quantum current commutation relation in [35] is given in the multiplicative form, L ( x ) R ( x e − hc /x ) L ( x ) R ( x /x ) − = R ( x /x ) − L ( x ) R ( x e − hc /x ) L ( x ) . (0.2)Its significance comes from Ding’s quantum current realization of the quantum affinealgebra U h ( b g N ) [6], which relies on the famous Ding–Frenkel isomorphism [7]. The algebragenerators are given as coefficients of matrix entries of the quantum current L ( x ), sothat L ( x ) belongs to End C N ⊗ U h ( b g N )[[ x ± ]], while the defining relations at the level c ∈ C are given by (0.2), along with one more family of relations in the g N = sl N case. As in [31, Sect. 5], in this paper we consider the φ -coordinated V c ( g N )-modules forthe associate φ ( z , z ) = z e z , which connects commutation relations (0.1) and (0.2).More specifically, by applying the substitutions x i = ze u i with i = 1 ,
2, multiplicativerelation (0.2) takes the additive form as in (0.1). It is worth noting that both additiveand multiplicative forms of the trigonometric R -matrix naturally occur in the theories ofquantum groups and exactly solvable models; see [12, 21, 34].As with the rational R -matrix case [27], the multiple copies of quantum currents L ( x i ) with i = 1 , . . . , n can be organized into the operators L [ n ] ( x , . . . , x n ) in the vari-ables x , . . . , x n which satisfy certain generalized version of commutation relation (0.2). We explain the precise meaning of relations (0.1) and (0.2) in Subsection 1.2. oughly speaking, such operators take place of the normal-ordered products of n quan-tum currents. In particular, for any restricted U h ( b g N )-module, i.e. for any module W suchthat L ( x ) w belongs to End C N ⊗ W (( x ))[[ h ]] for all w ∈ W , the series L [ n ] ( x , . . . , x n ) w possesses only finitely many negative powers of the variables x , . . . , x n modulo h k for all k > w ∈ W . By combining Ding’s quantum current realization [6] with Li’s theoryof φ -coordinated modules [31] and Cherednik’s fusion procedure for the trigonometric R -matrix [4] in the g N = sl N case, we establish the following correspondence betweenrestricted modules for the quantum affine algebra and φ -coordinated modules for theEtingof–Kazhdan quantum vertex algebra, which is the main result of this paper. Main Theorem.
Let g N = gl N , sl N . Let W be a restricted U h ( b g N ) -module of level c ∈ C .There exists a unique structure of φ -coordinated V c ( g N ) -module on W , where φ ( z , z ) = z e z , such that Y W ( T +[ n ] ( u , . . . , u n ) , z ) = L [ n ] ( x , . . . , x n ) (cid:12)(cid:12) x = ze u ,...,x n = ze un for all n > . (0.3) Conversely, let ( W, Y W ) be a φ -coordinated V c ( g N ) -module, where φ ( z , z ) = z e z . Thereexists a unique structure of restricted U h ( b g N ) -module of level c on W such that L ( z ) = Y W ( T + (0) , z ) . (0.4) Moreover, a topologically free C [[ h ]] -submodule W of W is a φ -coordinated V c ( g N ) -submodule of W if and only if W is an U h ( b g N ) -submodule of W . In order to establish this correspondence, some minor modifications had to be madeto the definitions of quantum affine algebra and φ -coordinated module. More specifically,both notions were redefined over the ring C [[ h ]] and suitably completed, so that they arecompatible with Etingof–Kazhdan’s definition of quantum vertex algebra.In the end, we recollect that the universal affine vertex algebra, which governs the repre-sentation theory of the corresponding affine Lie algebra b g N , is constructed on the vacuummodule over the universal enveloping algebra U( b g N ); see [20, 33]. In contrast, V c ( g N ) isnot the vacuum module over U h ( b g N ), although its quantum vertex algebra structure turnsinto the corresponding affine vertex algebra in the classical limit. Furthermore, it is notclear whether the vacuum module V c ( g N ) at the level c over the quantum affine algebraU h ( b g N ) possesses any natural quantum vertex algebra-like structure that governs the rep-resentation theory of U h ( b g N ). However, we have the following simple consequence of theMain Theorem: Corollary 0.1.
Let g N = gl N , sl N . The vacuum module V c ( g N ) over the quantum affinealgebra U h ( b g N ) is a φ -coordinated V c ( g N ) -module. Moreover, V c ( g N ) is an irreducible U h ( b g N ) -module if and only if it is an irreducible φ -coordinated V c ( g N ) -module. The paper is organized as follows. In Sections 1 and 2, we introduce the notationand provide preliminary definitions and results on restricted U h ( b g N )-modules and on φ -coordinated V c ( g N )-modules respectively. In Section 3, we prove the Main Theorem.Finally, in Section 4, we discuss a connection between the families of central elementsof the quantum affine algebra and the quantum affine vertex algebra established by φ -coordinated module map (0.3).1. Restricted modules for the quantum affine algebra
In this section, we recall some basic properties of the trigonometric R -matrix of type A . Next, we recall Ding’s quantum current realization of the quantum affine algebrain type A and the corresponding notion of restricted module. Also, we derive certainproperties of the quantum currents which are required in the following sections. Finally,we demonstrate how the Main Theorem implies Corollary 0.1. .1. Trigonometric R -matrix. We use the standard tensor notation, i.e. for any A = N X i,j,k,l =1 a ijkl e ij ⊗ e kl ∈ End C N ⊗ End C N and indices r, s = 1 , . . . , m such that r = s , where m > e ij ∈ End C N are thematrix units, we denote by A rs the element of the algebra (End C N ) ⊗ m , A rs = N X i,j,k,l =1 a ijkl ( e ij ) r ( e kl ) s , where ( e ij ) p = 1 ⊗ ( p − ⊗ e ij ⊗ ⊗ ( m − p ) . (1.1)Let N > h a formal parameter. Introduce the trigonometric R -matrix of type A by R ( x ) = N X i =1 e ii ⊗ e ii + e − h/ − x − e − h x N X i,j =1 i = j e ii ⊗ e jj + (cid:0) − e − h (cid:1) x − e − h x N X i,j =1 i>j e ij ⊗ e ji + 1 − e − h − e − h x N X i,j =1 i
2; see [19].Express the R -matrix R ( x ) defined by (1.12) as R ( x ) = g ( x ) R + ( x ) , where g ( x ) = f ( x )1 − e − h x , R + ( x ) = (cid:0) − e − h x (cid:1) R ( x ) . (1.15)Clearly, R + ( x ) is a polynomial with respect to the variable x , i.e. R + ( x ) belongs to(End C N ) ⊗ [[ h ]][ x ]. On the other hand, as ( e ah − x ∈ xh C [[ h ]], we conclude by (1.6)and (1.10) that g ( x ) admits the presentation g ( x ) = ∞ X k =0 g k x k (1 − x ) k +1 , where g k ∈ h k C [[ h ]] and g = 1 . (1.16)Denote by C ∗ ( z , . . . , z n ) the localization of the ring of Taylor series C [[ z , . . . , z n ]]at C [ z , . . . , z n ] × . Consider the unique embedding C ∗ ( z , . . . , z n ) → C (( z )) . . . (( z n )).Extending the embedding to the h -adic completion of C ∗ ( z , . . . , z n ) we obtain the map ι z ,...,z n : C ∗ ( z , . . . , z n )[[ h ]] → C (( z )) . . . (( z n ))[[ h ]] . (1.17)As in [26], we now apply the substitution x = e u to the normalized R -matrix R ( x ) givenby (1.12). First, replacing the variable x by e u in (1.16) we obtain g ( e u ) = ∞ X k =0 g k e ku (1 − e u ) k +1 = ∞ X k =0 g k e ku (cid:0) u − e u (cid:1) k +1 u k +1 ∈ C ∗ ( u )[[ h ]]since all numerators e ku u k +1 (1 − e u ) − k − belong to C [[ u ]] and g k ∈ h k C [[ h ]]. By applyingthe embedding ι u we get ι u g ( e u ) ∈ C (( u ))[[ h ]]. Next, as R + ( x ) is a polynomial withrespect to the variable x , by applying the substitution x = e u we obtain R + ( e u ), whichbelongs to (End C N ) ⊗ [[ h, u ]]. Finally, there exists a unique ψ ∈ h C [[ h ]] such that the R -matrix R ( e u ) := ψ ι u g ( e u ) R + ( e u ) ∈ End C N ⊗ End C N (( u ))[[ h ]] (1.18)possesses the unitarity property R ( e u ) R ( e − u ) = 1 (1.19)and the crossing symmetry properties R ( e u + Nh ) t D ( R ( e u ) − ) t = D and ( R ( e u ) − ) t D R ( e u + Nh ) t = D ; (1.20) ee [10, Prop. 1.2] and [26, Prop. 2.1]. Of course, R -matrix (1.18) also satisfies the Yang–Baxter equation R ( e u ) R ( e u + v ) R ( e v ) = R ( e v ) R ( e u + v ) R ( e u ) . (1.21)In what follows, whenever it is clear from the context, we omit the embedding symbol ι and write, e.g., f ( e u ) instead of ι u f ( e u ). Furthermore, in the multiple variable case,we employ the usual expansion convention where the choice of the embedding is deter-mined by the order of the variables. For example, if σ is a permutation in the symmetricgroup S n , then f ( e u σ + ... + u σn ) denotes ι u σ ,...,u σn f ( e u σ + ... + u σn ) ∈ C (( u σ )) . . . (( u σ n ))[[ h ]].In particular, by R ( e u + v ) in (1.21) is denoted ι u,v g ( e u + v ) R +13 ( e u + v ).1.2. Quantum affine algebra.
Ding’s quantum current realization of the quantumaffine algebra of type A was given in [6, Prop. 3.1]. We slightly modify the originaldefinition [6, Def. 3.1] in order to make the setting compatible with the quantum ver-tex algebra theory; see Remark 1.3 for more details. Our exposition starts in parallelwith [27, Subsection 2.1], where a certain quantum current algebra associated with thesuitably normalized Yang R -matrix was introduced. We omit some simple proofs as theypresent a straightforward generalization of the arguments from the aforementioned paperto the trigonometric case.For any integer N > N ) the associative algebra over the ring C [[ h ]]generated by the elements 1, C and λ ( r ) ij , where i, j = 1 , . . . , N and r ∈ Z , subject to thedefining relations C · a = a · C and 1 · a = a · a for all a ∈ F( N ) , i.e. 1 is the unit and C is a central element in F( N ). Introduce the Laurent series λ ij ( x ) = δ ij − h X r ∈ Z λ ( r ) ij x − r − ∈ F( N )[[ x ± ]] , where i, j = 1 , . . . , N, (1.22)and arrange them into the matrix L ( x ) ∈ End C N ⊗ F( N )[[ x ± ]], L ( x ) = N X i,j =1 e ij ⊗ λ ij ( x ) . (1.23)We now introduce certain completion of the algebra F( N ) which is suitable for express-ing the defining relations for the quantum affine algebra. For an integer p > p ( N )be the left ideal in F( N ) generated by all λ ( r ) ij , where i, j = 1 , . . . , N and r > p −
1. Definethe completion of F( N ) as the inverse limit e F( N ) = lim ←− F( N ) / I p ( N ) . The algebra e F( N ) is naturally equipped with the h -adic topology and its h -adic comple-tion is equal to e F( N )[[ h ]]. For any integer p > hp ( N ) be the h -adically completedleft ideal in e F( N )[[ h ]] generated by I p ( N ) and the element h p · L ( x ) so that the subscriptindicates the copy in the corresponding tensor product algebra, L r ( x ) = N X i,j =1 ⊗ ( r − ⊗ e ij ⊗ ⊗ ( m − r ) ⊗ λ ij ( x ) ∈ (End C N ) ⊗ m ⊗ F( N )[[ x ± ]] . (1.24)Employing such notation for m = 2 and r = 1 , L (1)[2] ( x, y ) = L ( x ) R ( ye − hC /x ) L ( y ) R ( y/x ) − , L (2)[2] ( x, y ) = R ( x/y ) − L ( y ) R ( xe − hC /y ) L ( x ) . n accordance with the discussion in Subsection 1.1, the R -matrices R ( ye ahC /x ) ± and R ( xe ahC /y ) ± with a ∈ C are regarded as Taylor series with respect to y/x and x/y respectively. By arguing as in [27, Lemma 2.1], one can prove Lemma 1.1.
The expressions L (1)[2] ( x, y ) and L (2)[2] ( x, y ) are well-defined elements of End C N ⊗ End C N ⊗ e F( N )[[ x ± , y ± , h ]] . Moreover, for any integer p > both L (1)[2] ( x, y ) and L (2)[2] ( y, x ) modulo I hp ( N ) belong to End C N ⊗ End C N ⊗ F( N )[[ x ± ]](( y )) . By Lemma 1.1, there exist elements λ ( r,s ; t ) ij k l in e F( N )[[ h ]] such that L ( t )[2] ( x, y ) = N X i,j,k,l =1 X r,s ∈ Z e ij ⊗ e kl ⊗ λ ( r,s ; t ) ij k l x − r − y − s − for t = 1 , . Let J( N ) be the ideal in the algebra e F( N )[[ h ]] generated by all elements λ ( r,s ;1) ij k l − λ ( r,s ;2) ij k l , where r, s ∈ Z and i, j, k, l = 1 , . . . , N. (1.25)Introduce the completion of J( N ) as the inverse limit e J( N ) = lim ←− J( N ) / J( N ) ∩ I p ( N ) . Note that the h -adic completion [ e J( N )][[ h ]] of[ e J( N )] = n a ∈ e F( N )[[ h ]] : h n a ∈ e J( N ) for some integer n > o is also an ideal in e F( N )[[ h ]]. Following [6, Def. 3.1], we define the (completed) quantumaffine algebra U h ( b gl N ) as the quotient of the algebra e F( N )[[ h ]] by the ideal [ e J( N )][[ h ]],U h ( b gl N ) = e F( N )[[ h ]] / [ e J( N )][[ h ]] . (1.26)Denote the images of the elements 1, C and λ ( r ) ij in quotient (1.26) again by 1, C and λ ( r ) ij . Also, denote by λ ij ( x ) and L ( x ) the corresponding series in U h ( b gl N )[[ x ± ]] andEnd C N ⊗ U h ( b gl N )[[ x ± ]] respectively. Defining relations (1.25) for the algebra U h ( b gl N ) canbe expressed by the quantum current commutation relation L ( x ) R ( ye − hC /x ) L ( y ) R ( y/x ) − = R ( x/y ) − L ( y ) R ( xe − hC /y ) L ( x ) , (1.27)as given by Reshetikhin and Semenov-Tian-Shansky [35]. As the images of the elements λ ( r,s ;1) ij k l and λ ( r,s ;2) ij k l in quotient (1.26) coincide, we denote them by λ ( r,s ) ij k l . Also, we write L [2] ( x, y ) = N X i,j,k,l =1 X r,s ∈ Z e ij ⊗ e kl ⊗ λ ( r,s ) ij k l x − r − y − s − . (1.28)and L [1] ( x ) = L ( x ). Observe that the both sides of relation (1.27) coincide with L [2] ( x, y ).Motivated by [35], we refer to the series L ( x ) as quantum currents . Our next goal is toderive a certain generalized version of (1.27) consisting of n + m quantum currents.For integers n, m > z and thefamilies of variables x = ( x , . . . , x n ) and y = ( y , . . . , y m ) with values in the space(End C N ) ⊗ n ⊗ (End C N ) ⊗ m by R nm ( zxe ah /y ) = −→ Y i =1 ,...,n ←− Y j = n +1 ,...,n + m R ij ( zx i e ah /y j − n ) , (1.29) Notice the swapped variables in this term. nm ( ye ah /zx ) = ←− Y i =1 ,...,n −→ Y j = n +1 ,...,n + m R ji ( y j − n e ah /zx i ) , (1.30)where a ∈ C and the arrows indicate the order of the factors. For example, we have R ( zx/y ) = R R R R and R ( y/xz ) = R ′ R ′ R ′ R ′ , where R ij = R ij ( zx i /y j − n ) and R ′ ji = R ji ( y j − n /zx i ). The corresponding functions as-sociated with the R -matrix R + ( x ) given by (1.15), R +12 nm ( zxe ah /y ) and R +21 nm ( ye ah /zx ),can be defined analogously. Note that the evaluations of (1.29) and (1.30) at z = 1 arewell-defined. We denote them by R nm ( xe ah /y ) and R nm ( ye ah /x ) respectively. Next, forany integer n > x = ( x , . . . , x n ) define the functions withvalues in (End C N ) ⊗ n by R [ n,a ] ( x ) = −→ Y i =1 ,...,n − −→ Y j = i +1 ,...,n R ji ( x j e − ah /x i ) − , (1.31) ~R [ n,a ] ( x ) = ←− Y i =1 ,...,n − ←− Y j = i +1 ,...,n R ji ( x j e − ah /x i ) − , (1.32)where a ∈ C and the arrows again indicate the order of the factors. For example, we have R [ n,a ] ( x ) = R R R R R R and ~R [ n,a ] ( x ) = R R R R R R , where R ji = R ji ( x j e − ah /x i ) − . If a = 0, we omit the second subscript and write R [ n ] ( x ) = R [ n, ( x ) and ~R [ n ] ( x ) = ~R [ n, ( x ) . Finally, for any integer n > L [2] ( x, y ), as given by (1.28), by setting L [ n ] ( x ) = −→ Y i =1 ,...,n (cid:0) L i ( x i ) R i +1 i ( x i +1 e − hC /x i ) . . . R ni ( x n e − hC /x i ) (cid:1) · ~R [ n ] ( x ) . (1.33)Denote by I hp ( b gl N ) , I p ( b gl N ) the images of the left ideals I hp ( N ) , I p ( N ) ⊂ e F( N )[[ h ]] inthe algebra U h ( b gl N ) with respect to the canonical map e F( N )[[ h ]] → U h ( b gl N ). In the nextproposition, we use the superscripts 1 , , z }| { (End C N ) ⊗ n ⊗ z }| { (End C N ) ⊗ m ⊗ z }| { U h ( b gl N ) . The proposition can be proved by using Lemma 1.1, Yang–Baxter equation (1.3), quantumcurrent commutation relation (1.27) and arguing as in [27, Prop. 2.4 and 2.5].
Proposition 1.2.
For any integers n, m > and the families of variables x = ( x , . . . , x n ) and y = ( y , . . . , y m ) we have:(1) The expression L [ n ] ( x ) is a well-defined element of (End C N ) ⊗ n ⊗ U h ( b gl N )[[ x ± , . . . , x ± n ]] . (2) For any p > the element L [ n ] ( x ) modulo I hp ( b gl N ) belongs to (End C N ) ⊗ n ⊗ U h ( b gl N )(( x , . . . , x n )) . (3) The following quantum current commutation relation holds: L n ] ( x ) R nm ( ye − hC /x ) L m ] ( y ) R nm ( y/x ) − = R nm ( x/y ) − L m ] ( y ) R nm ( xe − hC /y ) L n ] ( x ) . (1.34) Moreover, both sides of (1.34) coincide with L [ n + m ] ( x, y ) . eneralizing (1.28) we denote the coefficients of the matrix entries in (1.33) as follows: L [ n ] ( x ) = N X i ,j ,...,i n ,j n =1 X r ,...,r n ∈ Z e i j ⊗ . . . ⊗ e i n j n ⊗ λ ( r ,...,r n ) i j ...i n j n x − r − . . . x − r n − n . Our next goal is to introduce the quantum affine algebra associated with the affine Liealgebra b sl N . Let P h be the h -permutation operator, P h = N X i =1 e ii ⊗ e ii + e h/ N X i,j =1 i>j e ij ⊗ e ji + e − h/ N X i,j =1 i 1, where σ i is the transposition ( i, i + 1). For a reduceddecomposition of a permutation σ = σ i . . . σ i k ∈ S n set P hσ = P hσ i . . . P hσ ik . Let A ( n ) bethe image of the normalized anti-symmetrizer with respect to this action, so that A ( n ) = 1 n ! X σ ∈ S n sgn σ · P hσ . (1.35)Define the quantum determinant of the matrix L ( x ) byqdet L ( x ) = tr ,...,N A ( N ) L [ N ] ( x , . . . , x N ) (cid:12)(cid:12) x = x,...,x N = xe − ( N − h D . . . D N , (1.36)where the trace is taken over all N copies of End C N and the matrix D is given by (1.14).The quantum determinant is a formal power series in the variable x with coefficients in thequantum affine algebra, i.e. qdet L ( x ) belongs to U h ( b gl N )[[ x ± ]]. Indeed, the substitution x = x, . . . , x N = xe − ( N − h in (1.36) is well-defined due to the second assertion ofProposition 1.2. Furthermore, all coefficients d r of the quantum determinantqdet L ( x ) = 1 − h X r ∈ Z d r x r (1.37)belong to the center of the quantum affine algebra at the level c ∈ C ; see Proposition 4.1.Let I qdet be the ideal in the algebra U h ( b gl N ) generated by the elements d r , where r ∈ Z .Introduce its completion as the inverse limit e I qdet = lim ←− I qdet / I qdet ∩ I p ( gl N ) . The h -adic completion [ e I qdet ][[ h ]] of[ e I qdet ] = n a ∈ U h ( b gl N ) : h n a ∈ e I qdet for some integer n > o is also an ideal in U h ( b gl N ). Define the (completed) quantum affine algebra U h ( b sl N ) as thequotient of the algebra U h ( b gl N ) by the relation qdet L ( x ) = 1, i.e.U h ( b sl N ) = U h ( b gl N ) / [ e I qdet ][[ h ]] . Remark 1.3. In Ding’s definition [6, Def. 3.1], the quantum affine algebra is introducedas an associative algebra over the field C ( q ). However, as our goal is to study quan-tum vertex algebras associated to quantum affine algebras, we used the identification q = e h/ and introduced the quantum affine algebra as a suitably completed associativealgebra over the commutative ring C [[ h ]]. Thus we established the setting compatiblewith Etingof–Kazhdan’s notion of quantum vertex algebra [11, Sect. 1.4], which, in par-ticular, is required to be a topologically free C [[ h ]]-module; see also Li’s notion of h -adicquantum vertex algebra [30, Def. 2.20]. Furthermore, in contrast with Ding’s realization,we use normalized R -matrix (1.12) instead of (1.2). Such choice of the R -matrix enables he constructions of certain large families of central elements of the quantum affine alge-bra at the critical level and of the topological generators of the quantum Feigin–Frenkelcenter, as demonstrated in [13] and [26] respectively; see also Section 4.1.3. Restricted modules. Recall that a C [[ h ]]-module W is said to be torsion-free if hw = 0 for all nonzero w ∈ W and that W is said to be separable if ∩ n > h n W = 0.Moreover, W is said to be topologically free if it is separable, torsion-free and completewith respect to h -adic topology; see [25, Chapter XVI].Let g N = gl N , sl N . By arguing as in [27, Prop. 2.2] one can show that the algebraU h ( b g N ) is topologically free. Define a restricted U h ( b g N )-module W as a topologically free C [[ h ]]-module such that L ( x ) w ∈ End C N ⊗ W (( x ))[[ h ]] for all w ∈ W. (1.38) Proposition 1.4. Let W be a restricted U h ( b g N ) -module. Then for any n > and thevariables x = ( x , . . . , x n ) we have L [ n ] ( x ) w ∈ (End C N ) ⊗ n ⊗ W (( x , . . . , x n ))[[ h ]] for all w ∈ W. (1.39) Proof. Apply quantum current commutation relation (1.27) on an arbitrary element ofsome restricted module. For every integer k > y modulo h k while the right hand side contains finitelymany negative powers of the variable x modulo h k . Hence the statement of the propositionholds for n = 2. The case n > n which relies on (1.34). (cid:3) Remark 1.5. Note that (1.39) implies L [ n ] ( x ) ∈ End C N ⊗ Hom( W, W (( x , . . . , x n ))[[ h ]])for all n > 1. Hence we can apply the substitutions x = ze u , . . . , x n = ze u n , thus getting L [ n ] ( x , . . . , x n ) (cid:12)(cid:12) x = ze u ,...,x n = ze un ∈ End C N ⊗ Hom( W, W (( z ))[[ h, u , . . . , u n ]]) . (1.40)We will often denote the expression in (1.40) more briefly by L [ n ] ( x ) | x i = ze ui .As usual, an U h ( b g N )-module W is said to be of level c if the central element C ∈ U h ( b g N )acts on W as a scalar multiplication by some c ∈ C . Denote by U h ( b g N ) c the quantumaffine algebra at the level c , i.e. the quotient of U h ( b g N ) by the ideal generated by theelement C − c . Let K c be the left ideal in the algebra U h ( b g N ) c generated by all elements λ ( r ,...,r n ) i j ...i n j n such that r k > k = 1 , . . . , n, where n > i , . . . , i n , j , . . . , j n = 1 , . . . , N and r , . . . , r n ∈ Z . Introduce the completionof K c as the inverse limit e K c = lim ←− K c / K c ∩ I p ( b g N ) . Then the h -adic completion [ e K c ][[ h ]] of[ e K c ] = n a ∈ U h ( b g N ) c : h n a ∈ e K c for some n > o is also a left ideal in U h ( b g N ) c . Define the vacuum module V c ( g N ) at the level c over thequantum affine algebra U h ( b g N ) as the quotient of U h ( b g N ) c by its left ideal [ e K c ][[ h ]], V c ( g N ) = U h ( b g N ) c / [ e K c ][[ h ]] . (1.41)Observe that the canonical map U h ( b g N ) c → V c ( g N ) maps the left ideal I hp ( b g N ) to h p V c ( g N ). Denote by the image of the unit 1 ∈ U h ( b g N ) with respect to this map. Proposition 1.6. The vacuum module V c ( g N ) is a topologically free C [[ h ]] -module. More-over, it is a restricted U h ( b g N ) -module. roof. The first assertion is verified by arguing as in [27, Prop. 2.2]. As for the secondassertion, we first observe that all elements λ ( r ,...,r n ) i j ...i n j n such that n > r k < k = 1 , . . . , n (1.42)span an h -adically dense C [[ h ]]-submodule of V c ( g N ). Indeed, this follows from the factthat each monomial λ ( s ) k l . . . λ ( s m ) k m l m ∈ V c ( g N ) can be expressed using elements (1.42).This is done by employing crossing symmetry properties (1.13) and invertibility of thetrigonometric R -matrix to move all R -matrices which appear on the right hand side of L [ a + b ] ( x, y ) = L a ] ( x ) R ab ( ye − hC /x ) L b ] ( y ) R ab ( y/x ) − , (1.43)where a + b = m , x = ( x , . . . , x a ) and y = ( y , . . . , y b ), to the left hand side (for more de-tails see Remark 2.4), and then taking the coefficient of x − s − . . . x − s a − a y − s a +1 − . . . y − s m − b at the matrix entry e k l ⊗ . . . ⊗ e k m l m . Note that (1.43) follows from Proposition 1.2.Therefore, it is sufficient to check that L ( z ) w belongs to End C N ⊗ V c ( g N )(( z ))[[ h ]] forall w ∈ V c ( g N ) of the form as in (1.42). However, as (1.43) contains only nonnegativepowers of the variables x , . . . , x a , y , . . . , y b , this follows by setting a = 1 and b = n in(1.43), then moving R n ( ye − hC /x ) and R n ( y/x ) − to the left hand side and, finally, bytaking the coefficient of y − r − . . . y − r n − n at the matrix entries e ij ⊗ e i j ⊗ . . . ⊗ e i n j n for i, j = 1 , . . . , N . (cid:3) Observe that Proposition 1.6 and the Main Theorem imply Corollary 0.1.2. φ -Coordinated modules for the quantum affine vertex algebra In this section, we recall Etingof–Kazhdan’s construction of the quantum affine vertexalgebra associated with trigonometric R -matrix in type A . Next, we suitably modify Li’sdefinition of φ -coordinated module, thus establishing the setting for the Main Theorem.2.1. Quantum affine vertex algebra. We follow [9, 10] to introduce the R -matrixalgebras U + h ( b g N ); see also [12, 35]. Let U + h ( b gl N ) be the associative algebra over the ring C [[ h ]] generated by elements t ( − r ) ij , where i, j = 1 , . . . , N and r = 1 , , . . . , subject to thedefining relations R ( e u − v ) T +1 ( u ) T +2 ( v ) = T +2 ( v ) T +1 ( u ) R ( e u − v ) , (2.1)where T + ( u ) ∈ End C N ⊗ U + h ( b gl N )[[ u ]] is given by T + ( u ) = N X i,j =1 e ij ⊗ t + ij ( u ) for t + ij ( u ) = δ ij − h ∞ X r =1 t ( − r ) ij u r − ∈ U + h ( b gl N )[[ u ]] . As in (1.24), we use subscripts in (2.1) to indicate copies in the tensor product algebraEnd C N ⊗ End C N ⊗ U + h ( b gl N ). Note that the R -matrix R ( e u − v ) in defining relation (2.1)can be replaced by R + ( e u − v ).Define the quantum determinant of the matrix T + ( u ) byqdet T + ( u ) = tr ,...,N A ( N ) T +1 ( u ) . . . T + N ( u − ( N − h ) D . . . D N , (2.2)where the trace is taken over all N copies of End C N and the matrix D is given by (1.14).The quantum determinant qdet T + ( u ) belongs to U + h ( b gl N )[[ u ]]. Moreover, its coefficients δ r , which are given by qdet T + ( u ) = 1 − h X r > δ r u r , (2.3) elong to the center of the algebra U + h ( b gl N ); see proof of [26, Prop. 3.10]. Define thealgebra U + h ( b sl N ) as the quotient of U + h ( b gl N ) over the h -adically completed ideal generatedby the elements δ , δ , . . . Hence we have the following relation in U + h ( b sl N ):qdet T + ( u ) = 1 . (2.4)Let g N = gl N , sl N . For positive integers n and m we extend the notation in (1.29)and (1.30) by introducing the functions depending on the variable z and the families ofvariables u = ( u , ..., u n ) and v = ( v , ..., v m ) with values in the space (End C N ) ⊗ n ⊗ (End C N ) ⊗ m by R nm ( e z + u − v + ah ) = −→ Y i =1 ,...,n ←− Y j = n +1 ,...,n + m R ij ( e z + u i − v j − n + ah ) , (2.5) R nm ( e z + u − v + ah ) = ←− Y i =1 ,...,n −→ Y j = n +1 ,...,n + m R ji ( e z + u i − v j − n + ah ) , (2.6)where a ∈ C . Note that the expansion convention, as introduced at the end of Subsection1.1, is applied on every factor on the right hand side, i.e. R ij ( e z + u i − v j − n + ah ) = ψ ι z,u i ,v j − n g ( e z + u i − v j − n + ah ) R + ij ( e z + u i − v j − n + ah ) . If the variable z is omitted in (2.5) or (2.6), the embeddings ι u i ,v j − n are applied on the cor-responding normalizing functions g ( e u i − v j − n + ah ) instead. The functions R +12 nm ( e z + u − v + ah )and R +21 nm ( e z + u − v + ah ) corresponding to the R -matrix R + ( x ) given by (1.15) can be definedanalogously. Denote by the unit in the algebra U + h ( b g N ). We recall [11, Lemma 2.1]: Lemma 2.1. For any c ∈ C there exists a unique operator series T ∗ ( u ) ∈ End C N ⊗ Hom(U + h ( b g N )[[ h ]] , U + h ( b g N )(( u ))[[ h ]]) such that for all n > we have T ∗ ( u ) T +2 ( v ) . . . T + n +1 ( v n ) = R n ( e u − v + hc/ ) − T +2 ( v ) . . . T + n +1 ( v n ) R n ( e u − v − hc/ ) . (2.7)In order to indicate action (2.7), which is uniquely determined by the scalar c ∈ C ,we denote the topologically free C [[ h ]]-module U + h ( b g N )[[ h ]] by V c ( g N ). Following [11], weintroduce the operators on (End C N ) ⊗ n ⊗ V c ( g N ) by T +[ n ] ( u | z ) = T +1 ( z + u ) . . . T + n ( z + u n ) and T ∗ [ n ] ( u | z ) = T ∗ ( z + u ) . . . T ∗ n ( z + u n ) . By the expansion convention from Subsection 1.1, the operator T ∗ [ n ] ( u | z ) contains onlynonnegative powers of the variables u , . . . , u n as the embeddings ι z,u i are applied on itscorresponding factors. If the variable z is omitted, we write T +[ n ] ( u ) = T +1 ( u ) . . . T + n ( u n ) and T ∗ [ n ] ( u ) = T ∗ ( u ) . . . T ∗ n ( u n ) . (2.8)The next proposition, as given in [11, Prop. 2.2], is verified using (2.1) and (2.7). Inrelations (2.9)–(2.11), the superscripts 1 , , z }| { (End C N ) ⊗ n ⊗ z }| { (End C N ) ⊗ m ⊗ z }| { V c ( g N ) . For example, the superscripts 1 , T ∗ n ] ( u | z ) indicate that the operator T ∗ [ n ] ( u | z ) isapplied on the tensor factors 1 , . . . , n and n + m + 1. roposition 2.2. For any integers n, m > and the families of variables u = ( u , . . . , u n ) and v = ( v , . . . , v m ) the following equalities hold on V c ( g N ) : R nm ( e z − z + u − v ) T ∗ n ] ( u | z ) T ∗ m ] ( v | z ) = T ∗ m ] ( v | z ) T ∗ n ] ( u | z ) R nm ( e z − z + u − v ) , (2.9) R nm ( e z − z + u − v ) T +13[ n ] ( u | z ) T +23[ m ] ( v | z ) = T +23[ m ] ( v | z ) T +13[ n ] ( u | z ) R nm ( e z − z + u − v ) , (2.10) R nm ( e z − z + u − v + hc/ ) T ∗ n ] ( u | z ) T +23[ m ] ( v | z ) = T +23[ m ] ( v | z ) T ∗ n ] ( u | z ) R nm ( e z − z + u − v − hc/ ) . (2.11)From now on, the tensor products are understood as h -adically completed. The notionof quantum vertex algebra was introduced by Etingof and Kazhdan [11]. It is defined asa quadruple ( V, Y, , S ) such that1. V is a topologically free C [[ h ]]-module.2. Y = Y ( z ) is the vertex operator map , i.e. a C [[ h ]]-module map Y : V ⊗ V → V (( z ))[[ h ]] u ⊗ v Y ( z )( u ⊗ v ) = Y ( u, z ) v = X r ∈ Z u r v z − r − which satisfies the weak associativity : for any u, v, w ∈ V and n ∈ Z > there exists p ∈ Z > such that( z + z ) p Y ( u, z + z ) Y ( v, z ) w − ( z + z ) p Y (cid:0) Y ( u, z ) v, z (cid:1) w ∈ h n V [[ z ± , z ± ]] . (2.12)3. is the vacuum vector , i.e. a distinct element of V satisfying Y ( , z ) v = v, Y ( v, z ) ∈ V [[ z ]] and lim z → Y ( v, z ) = v for all v ∈ V, (2.13)4. S = S ( z ) is the braiding map , i.e. a C [[ h ]]-module map V ⊗ V → V ⊗ V ⊗ C (( z ))[[ h ]]which satisfies the S - locality : for any u, v ∈ V and n ∈ Z > there exists p ∈ Z > such thatfor all w ∈ V ( z − z ) p Y ( z ) (cid:0) ⊗ Y ( z ) (cid:1)(cid:0) S ( z − z )( u ⊗ v ) ⊗ w (cid:1) − ( z − z ) p Y ( z ) (cid:0) ⊗ Y ( z ) (cid:1) ( v ⊗ u ⊗ w ) ∈ h n V [[ z ± , z ± ]] . (2.14)The given data should posses several other properties which we omit as they are notused in this paper; for a complete definition see [11, Sect. 1.4]. Finally, we recall Etingof–Kazhdan’s construction [11, Thm. 2.3] in the trigonometric R -matrix case: Theorem 2.3. For any c ∈ C there exists a unique quantum vertex algebra structure on V c ( g N ) such that the vertex operator map Y is given by Y (cid:0) T +[ n ] ( u ) , z (cid:1) = T +[ n ] ( u | z ) T ∗ [ n ] ( u | z + hc/ − , (2.15) the vacuum vector is ∈ V c ( g N ) and the braiding map S ( z ) is defined by the relation S ( z ) (cid:0) R nm ( e z + u − v ) − T +24[ m ] ( v ) R nm ( e z + u − v − hc ) T +13[ n ] ( u )( ⊗ ) (cid:1) = T +13[ n ] ( u ) R nm ( e z + u − v + hc ) − T +24[ m ] ( v ) R nm ( e z + u − v )( ⊗ ) (2.16) for operators on (End C N ) ⊗ n ⊗ (End C N ) ⊗ m ⊗ V c ( g N ) ⊗ V c ( g N ) . Remark 2.4. Crossing symmetry properties (1.20) of R -matrix (1.18) can be expressedusing the ordered product notation as( D R ( e u + hN ) D − ) · RL R ( e u ) − = 1 and ( D R ( e u ) − D − ) · LR R ( e u + hN ) = 1 , (2.17)where the subscript RL (LR) indicates that the first tensor factor of D i R ( e u ) − D − i , i = 1 , 2, is applied from the right (left) while the second tensor factor is applied from the eft (right). Such notation naturally extends to the products of multiple R -matrices suchas (2.5) and (2.6). For example, by (2.17), we have (cid:0) D m ] R nm ( e z + u − v − h ( N + c ) ) − ( D m ] ) − (cid:1) · LR R nm ( e z + u − v − hc ) = 1 , (2.18)where D m ] = 1 ⊗ n ⊗ D ⊗ m and the subscript LR now indicates that the tensor factors1 , . . . , n ( n + 1 , . . . , n + m ) are applied from the left (right). As with (2.17), one canwrite crossing symmetry properties (1.13) of R -matrix (1.12) using the ordered productnotation. As before, the notation naturally extends to the multiple R -matrix productssuch as (1.29)–(1.32).Combining (2.16) and (2.18) we find the explicit formula for the action of the braiding, S ( z ) (cid:0) T +13[ n ] ( u ) T +24[ m ] ( v )( ⊗ ) (cid:1) = (cid:0) D m ] R nm ( e z + u − v − h ( N + c ) ) − ( D m ] ) − (cid:1) · LR (cid:0) R nm ( e z + u − v ) T +13[ n ] ( u ) R nm ( e z + u − v + hc ) − T +24[ m ] ( v ) R nm ( e z + u − v )( ⊗ ) (cid:1) . (2.19) Remark 2.5. As with (1.6), by formal Taylor Theorem (1.5) we have11 − xe u − v + ah = ∞ X k =0 ( e u − v + ah − k x k k ! ∂ k ∂x k (cid:18) − x (cid:19) . Therefore, due to (1.15), we can regard the R -matrix R ( xe u − v + ah ) as an element of(End C N ) ⊗ ( x )[[ u, v, h ]] for any a ∈ C , i.e. as a rational function in the variable x . Clearly,applying the embedding ι x,u,v we obtain an element of (End C N ) ⊗ (( x ))[[ u, v, h ]].We now extend the notation (2.5) and (2.6) by introducing the functions dependingon the variable x and the families of variables u = ( u , ..., u n ) and v = ( v , ..., v m ) withvalues in the space (End C N ) ⊗ n ⊗ (End C N ) ⊗ m by R nm ( xe u − v + ah ) = −→ Y i =1 ,...,n ←− Y j = n +1 ,...,n + m R ij ( xe u i − v j − n + ah ) , (2.20) R nm ( xe u − v + ah ) = ←− Y i =1 ,...,n −→ Y j = n +1 ,...,n + m R ji ( xe u i − v j − n + ah ) , (2.21)where a ∈ C . In accordance with Remark 2.5, the R -matrices in (2.20) and (2.21) areregarded as rational functions in the variable x . We use the map given by the followinglemma in Definition 2.7 below, to introduce the notion of φ -coordinated V c ( g N )-module. Lemma 2.6. There exists a unique C [[ h ]] -module map b S ( x ) : V c ( g N ) ⊗ V c ( g N ) → V c ( g N ) ⊗ V c ( g N )( x )[[ h ]] such that b S ( x ) (cid:0) T +13[ n ] ( u ) T +24[ m ] ( v )( ⊗ ) (cid:1) = (cid:0) D m ] R nm ( xe u − v − h ( N + c ) ) − ( D m ] ) − (cid:1) · LR (cid:0) R nm ( xe u − v ) T +13[ n ] ( u ) R nm ( xe u − v + hc ) − T +24[ m ] ( v ) R nm ( xe u − v )( ⊗ ) (cid:1) . (2.22) Moreover, the map b S ( x ) satisfies b S ( x ) (cid:0) R nm ( xe u − v ) − T +24[ m ] ( v ) R nm ( xe u − v − hc ) T +13[ n ] ( u )( ⊗ ) (cid:1) = T +13[ n ] ( u ) R nm ( xe u − v + hc ) − T +24[ m ] ( v ) R nm ( xe u − v )( ⊗ ) . (2.23) Proof. The map b S ( x ) is well-defined by (2.22), i.e. it maps the ideal of relations (2.1), and(2.4) in the g N = sl N case, to itself. Indeed, this follows by a straightforward calculationwhich relies on the identity R ( e u − v ) D D = D D R ( e u − v ) nd the following version of Yang–Baxter equation (1.3): R ( e u − v ) R ( xe u + αh ) R ( xe v + αh ) = R ( xe v + αh ) R ( xe u + αh ) R ( e u − v ) , α ∈ C . Moreover, the proof in the g N = sl N case employs identity (1.11) and some properties ofthe anti-symmetrizer A ( N ) , which are given by (3.64), (3.65) and A ( N ) T +1 ( u ) T +2 ( u − h ) . . . T + N ( u − ( N − h ) = T + N ( u − ( N − h ) . . . T +2 ( u − h ) T +1 ( u ) A ( N ) , see [26, Equality (3.12)]. As for relation (2.23), it follows from (2.22) and the equality (cid:0) D m ] R nm ( xe u − v − h ( N + c ) ) − ( D m ] ) − (cid:1) · LR R nm ( xe u − v − hc ) = 1 , which is verified by using crossing symmetry properties (1.13); recall Remark 2.4. (cid:3) φ -Coordinated modules. The notion of φ -coordinated module, where φ is anassociate of the one-dimensional additive formal group, was introduced by Li [31]. Asin [31, Sect. 5], throughout this paper we consider the associate φ ( z , z ) = z e z . (2.24)Before we proceed to the definition of φ -coordinated module, we introduce some notation.Let V be a topologically free C [[ h ]]-module and a , . . . , a n , k > A of Hom( V, V [[ z ± , z ± , u , . . . , u n ]]) can be expressed as A = B + u a C + . . . u a n n C n + h k C n +1 for some (2.25) B ∈ Hom( V, V (( z , z ))[[ u , . . . , u n , h ]]) , C , . . . , C n +1 ∈ Hom( V, V [[ z ± , z ± , u , . . . , u n ]]) . To indicate the fact that A possesses a decomposition as in (2.25), we write A ∈ Hom( V, V (( z , z ))[[ u , . . . , u n ]]) mod u a , . . . , u a n n , h k . (2.26)Note that the substitution B (cid:12)(cid:12) z = φ ( z ,z ) = ι z ,z ,u ,...,u n (cid:16) B ( z , z , u , . . . , u n ) (cid:12)(cid:12) z = φ ( z ,z ) (cid:17) (2.27)is well-defined even though the substitution A (cid:12)(cid:12) z = φ ( z ,z ) does not exist in general. In whatfollows, the substitution z = φ ( z , z ) is always understood as in (2.27), i.e. the givenexpression is expanded in nonnegative powers of the variable z . In order to simplify ournotation, we denote (2.27) as A (cid:12)(cid:12) mod u a ,...,u ann ,h k z = φ ( z ,z ) = A ( z , z , u , . . . , u n ) (cid:12)(cid:12) mod u a ,...,u ann ,h k z = φ ( z ,z ) . (2.28)The element B as in (2.25) is clearly unique modulo n X i =1 u a i i Hom( V, V [[ z ± , z ± , u , . . . , u n ]]) + h k Hom( V, V [[ z ± , z ± , u , . . . , u n ]]) . Let g N = gl N , sl N . The following definition of φ -coordinated V c ( g N )-module is basedon [31, Def. 3.4], which we slightly modify in order to make it compatible with Etingof–Kazhdan’s quantum vertex algebra theory; see Remark 2.9 for more details. Definition 2.7. A φ -coordinated V c ( g N ) -module is a pair ( W, Y W ) such that W is atopologically free C [[ h ]]-module and Y W = Y W ( z ) is a C [[ h ]]-module map Y W : V c ( g N ) ⊗ W → W (( z ))[[ h ]] u ⊗ w Y W ( z )( u ⊗ w ) = Y W ( u, z ) w = X r ∈ Z u r w z − r − hich satisfies Y W ( , z ) w = w for all w ∈ W ; the weak associativity : for any u, v ∈ V c ( g N )and k ∈ Z > there exists p ∈ Z > such that( z − z ) p Y W ( u, z ) Y W ( v, z ) ∈ Hom( W, W (( z , z ))) mod h k and (2.29) (cid:0) ( z − z ) p Y W ( u, z ) Y W ( v, z ) (cid:1)(cid:12)(cid:12) mod h k z = φ ( z ,z ) − ( φ ( z , z ) − z ) p Y W ( Y ( u, z ) v, z ) ∈ h k Hom( W, W [[ z ± , z ± ]]); (2.30)and the b S -locality : for any u, v ∈ V c ( g N ) and k ∈ Z > there exists p ∈ Z > such that( z − z ) p Y W ( z ) (cid:0) ⊗ Y W ( z ) (cid:1) ι z ,z (cid:0) b S ( z /z )( u ⊗ v ) ⊗ w (cid:1) (2.31) − ( z − z ) p Y W ( z ) (cid:0) ⊗ Y W ( z ) (cid:1) ( v ⊗ u ⊗ w ) ∈ h k W [[ z ± , z ± ]] for all w ∈ W. Let W be a topologically free C [[ h ]]-submodule of W . A pair ( W , Y W ) is said to be a φ -coordinated V c ( g N ) -submodule of W if Y W ( v, z ) w belongs to W for all v ∈ V c ( g N ) and w ∈ W , where Y W denotes the restriction and corestriction of Y W , Y W ( z ) = Y W ( z ) (cid:12)(cid:12)(cid:12) W (( z ))[[ h ]] V c ( g N ) ⊗ W : V c ( g N ) ⊗ W → W (( z ))[[ h ]] . Remark 2.8. Regarding the weak associativity, note that (2.29) and (2.30) employ thenotation introduced in (2.26) and (2.28) for n = 0, i.e. there are no variables u , . . . , u n .Next, observe that the b S -locality already implies that there exists p ∈ Z > such that(2.29) holds. However, we still include this requirement in the definition as it ensuresthat the integer p is large enough so that the substitution z = φ ( z , z ) in (2.30) iswell-defined. Finally, the motivation for expressing the weak associativity in the form asin (2.29) and (2.30) is given in [31, Rem. 3.2]. Remark 2.9. As with the quantum affine algebra in the previous section, we introducethe notion of φ -coordinated module over the ring C [[ h ]] instead of a field in order tomake it compatible with the Etingof–Kazhdan quantum vertex algebra theory; cf. originaldefinition [31, Def. 3.4]. Furthermore, unlike the original definition, we require that the φ -coordinated module map Y W ( z ) possesses b S -locality property (2.31). The general theorydeveloped by Li suggests that (2.31) might be omitted from the definition, due to the factthat the vertex operator map Y ( z ) already possesses S -locality property (2.14); see [31,Prop. 5.6]. However, we include the b S -locality in the definition in order to emphasizethe importance of quantum current commutation relation (1.27). More specifically, in theproof of the Main Theorem, we derive the b S -locality property directly from the quantumcurrent commutation relation; see Lemma 3.8.Introduce the series δ ( z ) = X k ∈ Z z k ∈ C [[ z ± ]] and log(1 + z ) = − ∞ X k =1 ( − z ) k k ∈ z C [[ z ]] . The following Jacobi-type identity was established in [31, Prop. 5.9]. Although, in contrastwith [31], we consider quantum vertex algebras and φ -coordinated modules defined overthe ring C [[ h ]], the next proposition can be proved by arguing as in the proofs of [31,Lemma 5.8] and [31, Prop. 5.9]. Proposition 2.10. Let W be a φ -coordinated V c ( g N ) -module, where φ ( z , z ) = z e z .For any u, v ∈ V c ( g N ) we have ( z z ) − δ (cid:18) z − z z z (cid:19) Y W ( z )(1 ⊗ Y W ( z ))( u ⊗ v ) (2.32) ( z z ) − δ (cid:18) z − z − z z (cid:19) Y W ( z )(1 ⊗ Y W ( z )) ι z ,z b S ( z /z )( v ⊗ u ) (2.33)= z − δ (cid:18) z (1 + z ) z (cid:19) Y W ( Y ( u, log(1 + z )) v, z ) . (2.34)3. Proof of the Main Theorem In this section we prove the Main Theorem. The proof is divided into four parts, Sub-sections 3.1–3.4. In Subsection 3.1, we obtain some properties of the normalizing functionsfor the trigonometric R -matrix which are required in the later stages of the proof; seeLemmas 3.1–3.4. In Subsection 3.2, we demonstrate how to establish the φ -coordinated V c ( gl N )-module structure on a restricted module of level c for the quantum affine algebraU h ( b gl N ); see Lemmas 3.5–3.8. The key ingredient in this part of the proof is Ding’s quan-tum current realization and, in particular, the fact that quantum current commutationrelation (1.27) resembles b S -locality property (2.31). In Subsection 3.3, we use Li’s Jacobi-type identity, as given in Proposition 2.10, to establish the structure of restricted moduleof level c for the quantum affine algebra U h ( b gl N ) on a φ -coordinated V c ( gl N )-module; seeLemma 3.9. Finally, we finish the proof in the g N = gl N case by showing that the C [[ h ]]-submodules invariant with respect to the action of the quantum affine algebra and withrespect to the corresponding action of the quantum vertex algebra coincide; see Lemma3.10. In Subsection 3.4, we use the fusion procedure for the two-parameter trigonometric R -matrix to extend the results to the g N = sl N case, thus completing the proof of theMain Theorem; see Lemmas 3.11–3.15.3.1. Normalizing functions. Introduce the function r ( x ) by r ( x ) = − xe h (1 − e h x ) − f ( x ) − , (3.1)where f ( x ) is given by (1.10). Lemma 3.1. The function r ( x ) ∈ C [[ x, h ]] satisfies R ( x ) − = r ( x ) R +12 (1 /x ) . (3.2) Moreover, it admits the presentation r ( x ) = ∞ X k =0 r k x k +1 (1 − x ) k +1 such that r k ∈ h k C [[ h ]] and r = − e h . (3.3) Proof. By combining unitarity property (1.4) and (1.15) we obtain R ( x ) − = (cid:0) f ( x ) R ( x ) (cid:1) − = f ( x ) − R ( x ) − = f ( x ) − R (1 /x )= f ( x ) − (1 − e − h x − ) − R +12 (1 /x ) = − xe h (1 − e h x ) − f ( x ) − R +12 (1 /x ) = r ( x ) R +12 (1 /x ) , as required. Next, by using (1.10) we find f ( x ) − = ∞ X l =0 − ∞ X k =1 f k (cid:18) x − x (cid:19) k ! l = 1 + ∞ X k =1 β k (cid:18) x − x (cid:19) k (3.4)for some β k ∈ h k C [[ h ]]. It is clear that the product of (1.6) for a = 1, (3.4) and − xe h isequal to r ( x ) and, furthermore, that it admits presentation (3.3). (cid:3) We use the following lemma in the proofs of weak associativity and b S -locality of the φ -coordinated module map, as well as to establish the restricted module structure on a φ -coordinated V c ( gl N )-module; see Lemmas 3.7, 3.8 and 3.9 respectively. emma 3.2. Let F = g ± or F = r ± . For any integers a , a , k > and α ∈ C thereexists an integer p > such that the coefficients of all monomials u a ′ u a ′ h k ′ , where a ′ < a , a ′ < a and k ′ < k, (3.5) in ( z − z ) p F ( z e u − u + αh /z ) belong to C [ z , z ± ] and such that the coefficients of allmonomials (3.5) in (cid:0) ( z − z ) p F ( z e u − u + αh /z ) (cid:1) (cid:12)(cid:12) mod u a ,u a ,h k z = z e z and z p ( e z − p F ( e z + u − u + αh ) (3.6) coincide. Proof. Set δ = 0 for F = g and δ = 1 for F = r , i.e. δ = δ F,r , so that we can considerboth cases simultaneously. Let U = C [[ x, x , h ]]. Recall (1.16) and (3.3). As the map ι x commutes with partial differential operator ∂/∂x , by using Taylor Theorem (1.5) we find ι x,x F ( x + x ) = ∞ X s =0 x l l ! ∂ l ∂x l ι x F ( x ) = ∞ X l,s =0 F s x l l ! ι x ∂ l ∂x l (cid:18) x s + δ (1 − x ) s +1 (cid:19) ∈ U, where F s = g s for F = g and F s = r s for F = r . By (1.16) and (3.3), every F s belongs to h s C [[ h ]], so all summands with s > k are trivial modulo h k U . Hence the given expressionmodulo U := x a + a + k − U + h k U contains only finitely many nonzero summands and,consequently, only finitely many terms (1 − x ) s +1 in the denominator. Therefore, thereexists an integer p > ι x,x (1 − x ) p F ( x + x ) = ∞ X l,s =0 F s x l l ! ι x (1 − x ) p ∂ l ∂x l (cid:18) x s + δ (1 − x ) s +1 (cid:19) ∈ C [ x, x , h ] mod U , where the equality holds modulo U and the map ι x can be omitted on the right handside as p can be chosen so that (1 − x ) p cancels all negative powers of (1 − x ) modulo U .By applying the substitution ( x, x ) = ( z /z , z ( e u − u + αh − /z ) to ι x,x (1 − x ) p F ( x + x ) mod U (3.7)and then multiplying the resulting expression by ( − z ) p we get( z − z ) p F ( z e u − u + αh /z ) ∈ C [ z , z ± , u , u , h ] mod V (3.8)for V = u a V + u a V + h k V and V = C [[ z , z ± , u , u , h ]], thus proving the first assertionof the lemma.Set W = u a W + u a W + h k W for W = C [[ z , z , u , u , h ]]. As (3.7) is a polynomialin the variables x and x , by applying the substitution ( x, x ) = ( e z , e z ( e u − u + αh − − z ) p we get z p ( e z − p F ( e z + u − u + αh ) mod W , (3.9)where, by the expansion convention from Subsection 1.1, F ( e z + u − u + αh ) stands for ι z ,u ,u F ( e z + u − u + αh ). Finally, as (3.8) modulo V is a polynomial with respect tothe variables z /z and z , by applying the substitution z = z e z we again obtain (3.9),thus proving the second assertion of the lemma.If F = g − or F = r − , one easily checks that F ( x ) = ∞ X s =0 F s x s − δ F,r − (1 − x ) s − for some F s ∈ h s C [[ h ]] , (3.10)so the lemma is verified by arguing as above. (cid:3) Note that the expression ι x,x (1 − x ) p F ( x + x ) is considered modulo U because, otherwise, theaforementioned substitution would not be well-defined (although the same substitution is well-definedwhen applied to (1 − x ) p F ( x + x ) with F ( x + x ) being regarded as a rational function with respect tothe variables x and x ). e now recall a certain useful consequence of [31, Lemma 2.7], as given in [31, Rem.2.8]: For any A ( z , z ) , B ( z , z ) ∈ C (( z , z )), the equality A ( z , z ) (cid:12)(cid:12) z = z e z = B ( z , z ) (cid:12)(cid:12) z = z e z implies A ( z , z ) = B ( z , z ) . (3.11)Since the C [[ h ]]-module C (( z , z ))[[ h ]] is separable, implication (3.11) clearly extends toany A ( z , z ) , B ( z , z ) ∈ C (( z , z ))[[ h ]]. Lemma 3.3. In C (( u ))[[ h ]] we have r ( e − u ) = ψ g ( e u ) . (3.12) Moreover, for any integers a , a , k > and α ∈ C there exists an integer p > such thatthe coefficients of all monomials (3.5) in ( z − z ) p r ( z e − u + u − αh /z ) and ( z − z ) p ψ g ( z e u − u + αh /z ) (3.13) coincide. Proof. By [26, Prop. 2.1] we have ψ f ( e u ) = f ( e − u ) − . Therefore, using (3.1) we get r ( e − u ) = − e − u + h (1 − e − u + h ) − f ( e − u ) − = − ψ e − u + h (1 − e − u + h ) − f ( e u )= ψ (1 − e u − h ) − f ( e u ) = ψ g ( e u ) , as required, where the last equality follows from (1.15). Next, by Lemma 3.2 and (3.12),there exists p > C [ z ± , z ± ] and such that the coefficients of all monomials (3.5) in (cid:0) ( z − z ) p r ( z e − u + u − αh /z ) (cid:1) (cid:12)(cid:12) mod u a ,u a ,h k z = z e z and (cid:0) ( z − z ) p ψ g ( z e u − u + αh /z ) (cid:1) (cid:12)(cid:12) mod u a ,u a ,h k z = z e z coincide. The second assertion of the lemma now follows by implication (3.11). (cid:3) The next lemma, which relies on Lemma 3.3, will be used in the proof of b S -locality ofthe φ -coordinated module map; see Lemma 3.8. Lemma 3.4. (1) Let F = g ± or F = r ± . There exists b F ( x, u, v ) in C ( x )[[ u, v, h ]] suchthat for all α ∈ C the following equality in C (( z ))[[ u, v, h ]] holds: b F ( e z , u, v − αh ) = F ( e z + u − v + αh ) . (3.14)(2) For any integers n, m > , the families of variables u = ( u , . . . , u n ) , v = ( v , . . . , v m ) and c ∈ C there exist functions b G ( x, u, v ) , b H ( x, u, v ) ∈ C ( x )[[ u , . . . , u n , v , . . . , v m , h ]] such that the following equalities hold in C (( z ))[[ u , . . . , u n , v , . . . , v m , h ]] : b G ( e z , u, v ) = G ( z, u, v ) and b H ( e z , u, v ) = H ( z, u, v ) , where G ( z, u, v ) = n Y i =1 m Y j =1 g ( e z + u i − v j − h ( N + c ) ) − g ( e z + u i − v j + hc ) − g ( e z + u i − v j ) , (3.15) H ( z, u, v ) = n Y i =1 m Y j =1 g ( e − z + u i − v j − h ( N + c ) ) − r ( e z − u i + v j − hc ) − g ( e − z + u i − v j ) r ( e z − u i + v j ) . (3.16)(3) Let a , . . . , a n , b , . . . , b m , k > be arbitrary integers and ι = ι z ,z ,u ,...,u n ,v ,...,v m theembedding. There exists an integer p > such that the coefficients of all monomials u a ′ . . . u a ′ n n v b ′ . . . v b ′ m m h k ′ , where a ′ i < a i , b ′ j < b j and k ′ < k in ( z − z ) p ι b G ( z /z , u, v ) and ( z − z ) p ι b H ( z /z , u, v ) coincide. roof. Due to (1.16), (3.3) and (3.10), we can regard g ( x ) ± and r ( x ) ± as elements of C ( x )[[ h ]]. Let F = g ± or F = r ± and write F ( x ) = P ∞ s =0 F s ( x ) h s for some F s ( x ) ∈ C ( x ).Applying formal Taylor Theorem (1.5) to z ι z F s ( e z ) we get for any α ∈ C ι z,u,v,h F s ( e z + u − v + αh ) = ∞ X l =0 ( u − v + αh ) l l ! ∂ l ∂z l ι z F s ( e z ) in C (( z ))[[ u, v, h ]] . The partial differential operator ∂/∂z commutes with the map ι z and all ∂ l ∂z l F s ( e z ) can benaturally regarded as elements of C ( e z ). Hence we can introduce functions b F l,s ( x ) ∈ C ( x )by the requirement b F l,s ( e z ) = ∂ l ∂z l F s ( e z ). The first statement of the lemma now clearlyfollows as the function b F ( x, u, v ) ∈ C ( x )[[ u, v, h ]] satisfying (3.14) can be defined by b F ( x, u, v ) = ∞ X s =0 ∞ X l =0 ( u − v ) l l ! b F l,s ( x ) ! h s . The second statement is proved by applying the first statement on each factor of (3.15)and (3.16). Finally, by (3.12) we have G ( z, u, v ) = H ( − z, u, v ), so the third statementfollows by Lemma 3.3. (cid:3) Establishing the φ -coordinated V c ( gl N ) -module structure. Let W be a re-stricted U h ( b gl N )-module of level c ∈ C . In this subsection, we prove the first assertion ofthe Main Theorem, i.e. we show that (0.3) defines a unique structure of φ -coordinated V c ( gl N )-module on W , where φ ( z , z ) = z e z . The proof is divided into four lemmaswhich verify all requirements imposed by Definition 2.7. Lemma 3.5. Formula (0.3) , together with Y W ( , z ) = 1 W , defines a unique C [[ h ]] -modulemap V c ( gl N ) ⊗ W → W (( z ))[[ h ]] . Proof. First, we note that the right hand side of (0.3) is well-defined, as was discussed inRemark 1.5. Next, we recall that the algebra U + h ( b gl N ) is spanned by all coefficients of allmatrix entries of T +[ n ] ( u ), n > 1, and ; see [9, Sect. 3.4] or [26, Prop. 2.4] . In order toprove the lemma, we have to show that v Y W ( v, z ) preserves the ideal of relations (2.1).More specifically, it is sufficient to check that for any integers n > i = 1 , , . . . , n − u = ( u , . . . , u n ) the expression R ii +1 ( e u i − u i +1 ) T +[ n ] ( u ) − P ii +1 T +[ n ] ( u i +1 ,i ) P ii +1 R ii +1 ( e u i − u i +1 ) , (3.17)where u i +1 ,i = ( u , . . . , u i − , u i +1 , u i , u i +2 , . . . , u n ), belongs to the kernel of v Y W ( v, z ).Let x = ( x , . . . , x n ) and x i +1 ,i = ( x , . . . , x i − , x i +1 , x i , x i +2 , . . . , x n ). Using Yang–Baxter equation (1.3) and commutation relation (1.27) one can prove the identity R ii +1 ( x i /x i +1 ) L [ n ] ( x ) = P ii +1 L [ n ] ( x i +1 ,i ) P ii +1 R ii +1 ( x i /x i +1 ) . (3.18)By Proposition 1.4, all matrix entries of L [ n ] ( x ) belong to Hom( W, W (( x , . . . , x n ))[[ h ]]),so all matrix entries in (3.18) belong toHom( W, W (( x i +1 ))(( x , . . . , x i , x i +2 , . . . , x n ))[[ h ]]) . Recall R -matrix decomposition (1.15). By (3.10) the function g ( x i /x i +1 ) − belongs to C [ x − i +1 ][[ h, x i ]], so we can multiply (3.18) by g ( x i /x i +1 ) − , thus getting R + ii +1 ( x i /x i +1 ) L [ n ] ( x ) = P ii +1 L [ n ] ( x i +1 ,i ) P ii +1 R + ii +1 ( x i /x i +1 ) . (3.19) We should mention that the notation in this paper slightly differs from [26]. In particular, the algebraU( R ), as defined in [26, Sect. 2], coincides with the algebra U + h ( b gl N ) defined in Subsection 2.1. ince the R -matrix R + ii +1 ( x i /x i +1 ) is a polynomial in x i /x i +1 , all matrix entries of bothsides in (3.19) belong to Hom( W, W (( x , . . . , x n ))[[ h ]]). Therefore, we can apply the sub-stitutions x i = ze u i with i = 1 , . . . , n to (3.19), thus getting the following equality in(End C N ) ⊗ n ⊗ Hom( W, W (( z ))[[ h, u , . . . , u n ]]): R + ii +1 ( e u i − u i +1 ) · (cid:0) L [ n ] ( x ) (cid:1) (cid:12)(cid:12) x i = ze ui = P ii +1 (cid:0) L [ n ] ( x i +1 ,i ) (cid:1) (cid:12)(cid:12) x i = ze ui P ii +1 R + ii +1 ( e u i − u i +1 ) . Multiplying the equality by ψg ( e u i − u i +1 ) ∈ C (( u i +1 ))[[ h, u i ]] and using (1.18) we find R ii +1 ( e u i − u i +1 ) (cid:0) L [ n ] ( x ) (cid:1) (cid:12)(cid:12) x i = ze ui − P ii +1 (cid:0) L [ n ] ( x i +1 ,i ) (cid:1) (cid:12)(cid:12) x i = ze ui P ii +1 R ii +1 ( e u i − u i +1 ) = 0 . As the left hand side coincides with the image of (3.17), with respect to Y W ( z ), weconclude that (0.3) defines a C [[ h ]]-module map V c ( gl N ) ⊗ W → W [[ z ± ]], as required.Moreover, by Remark 1.5 its image belongs to W (( z ))[[ h ]]. Finally, it is clear that the C [[ h ]]-module map Y W ( z ) is uniquely determined by (0.3). (cid:3) The next lemma follows from b S -locality property (2.31) which is verified in Lemma3.8 below; recall Remark 2.8. Nonetheless, we provide the direct proof as the underlyingcalculations are required in the proof of Lemma 3.7. Lemma 3.6. The map Y W ( z ) satisfies (2.29) , i.e. for any u, v ∈ V c ( gl N ) and k ∈ Z > there exists p ∈ Z > such that ( z − z ) p Y W ( u, z ) Y W ( v, z ) ∈ Hom( W, W (( z , z ))) mod h k . (3.20) Proof. For any integers n, m > u = ( u , . . . , u n ) and v =( v , . . . , v m ) we have Y W ( T +13[ n ] ( u ) , z ) Y W ( T +23[ m ] ( v ) , z ) = (cid:0) L n ] ( x ) (cid:1) (cid:12)(cid:12) x i = z e ui (cid:0) L m ] ( y ) (cid:1) (cid:12)(cid:12) y j = z e vj , (3.21)where x = ( x , . . . , x n ) and y = ( y , . . . , y m ). The coefficients in (3.21) are operators onthe multiple tensor product with superscripts 1 , , z }| { (End C N ) ⊗ n ⊗ z }| { (End C N ) ⊗ m ⊗ z}|{ W . Let us rewrite the right hand side in (3.21). The third assertion of Proposition 1.2 implies L n ] ( x ) R nm ( ye − hc /x ) L m ] ( y ) = L [ n + m ] ( x, y ) R nm ( y/x ) . (3.22)By expressing the second crossing symmetry relation in (1.13) in the variable x = y j e − h ( N + c ) /x i , then applying the transposition t and finally conjugating the resultingequality by the permutation operator P we find (cid:0) D R ( y j e − h ( N + c ) /x i ) − D − (cid:1) · RL R ( y j e − hc /x i ) = 1 . Furthermore, due to Lemma 3.1, we can write this equality as r ( y j e − h ( N + c ) /x i ) (cid:0) D R +12 ( x i e h ( N + c ) /y j ) D − (cid:1) · RL R ( y j e − hc /x i ) = 1 . Hence we have r ( x, y ) (cid:16) D n ] R +12 nm ( xe h ( N + c ) /y ) (cid:0) D n ] (cid:1) − (cid:17) · RL R nm ( ye − hc /x ) = 1 , where (3.23) D n ] = D ⊗ n ⊗ ⊗ m and r ( x, y ) = n Y i =1 m Y j =1 r ( y j e − h ( N + c ) /x i ) . (3.24)Using (3.23) we can move R nm ( ye − hc /x ) in (3.22) to the right hand side, which gives us L n ] ( x ) L m ] ( y ) = r ( x, y ) (cid:16) D n ] R +12 nm ( xe h ( N + c ) /y ) (cid:0) D n ] (cid:1) − (cid:17) · RL (cid:0) L [ n + m ] ( x, y ) R nm ( y/x ) (cid:1) r ( x, y ) g ( x, y ) (cid:16) D n ] R +12 nm ( xe h ( N + c ) /y ) (cid:0) D n ] (cid:1) − (cid:17) · RL (cid:0) L [ n + m ] ( x, y ) R +21 nm ( y/x ) (cid:1) , (3.25)where the second equality comes from (1.15) and the function g ( x, y ) is given by g ( x, y ) = n Y i =1 m Y j =1 g ( y j /x i ) . (3.26)Let a , . . . , a n , b , . . . , b m , k > x i = z e u i , y j = z e v j for i = 1 , . . . , n, j = 1 , . . . , m (3.27)to (3.25), thus getting (3.21), and then consider the coefficients of all monomials u a ′ . . . u a ′ n n v b ′ . . . v b ′ m m h k ′ , where 0 a ′ i < a i , b ′ j < b j and k ′ < k. (3.28)First, as the R -matrix R + ( w ) is a polynomial with respect to the variable w , we concludeby Proposition 1.4 and Remark 1.5 that (cid:16)(cid:16) D n ] R +12 nm ( xe h ( N + c ) /y ) (cid:0) D n ] (cid:1) − (cid:17) · RL (cid:0) L [ n + m ] ( x, y ) R +21 nm ( y/x ) (cid:1)(cid:17) (cid:12)(cid:12)(cid:12)(cid:12) x i = z e ui , y j = z e vj (3.29) ∈ (End C N ) ⊗ n ⊗ (End C N ) ⊗ m ⊗ Hom( W, W (( z , z ))[[ u , . . . , u n , v , . . . , v n , h ]]) . Next, by Lemma 3.2 there exists an integer p > 0, which depends on the choice of integers a , . . . , a n , b , . . . , b m , k , such that the coefficients of all monomials (3.28) in( z − z ) p (cid:0) r ( x, y ) g ( x, y ) (cid:1)(cid:12)(cid:12) x i = z e ui , y j = z e vj (3.30)belong to C [ z ± , z ± ]. Finally, we observe that the coefficients of all monomials (3.28) inthe product of (3.29) and (3.30) coincide with the corresponding coefficients in( z − z ) p Y W ( T +13[ n ] ( u ) , z ) Y W ( T +23[ m ] ( v ) , z ) . Therefore, by the preceding discussion, these coefficients belong to(End C N ) ⊗ n ⊗ (End C N ) ⊗ m ⊗ Hom( W, W (( z , z ))) , which implies the statement of the lemma. (cid:3) Lemma 3.7. The map Y W ( z ) satisfies weak associativity (2.30) , i.e. for any u, v ∈ V c ( gl N ) and k ∈ Z > there exists p ∈ Z > such that (3.20) holds and such that (cid:0) ( z − z ) p Y W ( u, z ) Y W ( v, z ) (cid:1)(cid:12)(cid:12) mod h k z = z e z − z p ( e z − p Y W ( Y ( u, z ) v, z ) ∈ h k Hom( W, W [[ z ± , z ± ]]) . (3.31) Proof. Let n, m, a , . . . , a n , b , . . . , b m , k > u = ( u , . . . , u n ) and v = ( v , . . . , v m ) the families of variables. Consider the coefficients of all monomials (3.28)in the expression (cid:0) ( z − z ) p Y W ( T +13[ n ] ( u ) , z ) Y W ( T +23[ m ] ( v ) , z ) (cid:1)(cid:12)(cid:12) mod u a ,...,u ann ,v b ,...,v bmm ,h k z = z e z , (3.32)which corresponds to the first summand in (3.31). As demonstrated in the proof of Lemma3.6, they coincide with the coefficients of all monomials (3.28) in the product (cid:18)(cid:16)(cid:16) D n ] R +12 nm ( xe h ( N + c ) /y ) (cid:0) D n ] (cid:1) − (cid:17) · RL (cid:0) L [ n + m ] ( x, y ) R +21 nm ( y/x ) (cid:1)(cid:17) (cid:12)(cid:12)(cid:12) x i = z e ui , y j = z e vj (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) z = z e z (3.33) × (cid:16) ( z − z ) p · (cid:0) r ( x, y ) g ( x, y ) (cid:1)(cid:12)(cid:12) x i = z e ui , y j = z e vj (cid:17) (cid:12)(cid:12)(cid:12) mod u a ,...,u ann ,v b ,...,v bmm ,h k z = z e z (3.34) or a suitably chosen integer p > a , . . . , a n , b , . . . , b m , k ). Recall thatthe functions r and g are given by (3.24) and (3.26). First, we observe that the coefficientsof all monomials (3.28) in factor (3.33) coincide with the corresponding coefficients in (cid:16) D n ] R +12 nm ( e z + u − v + h ( N + c ) ) (cid:0) D n ] (cid:1) − (cid:17) · RL (cid:18) L [ n + m ] ( x, y ) (cid:12)(cid:12)(cid:12) x i = z e ui y j = z e vj (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) z = z e z R +21 nm ( e − z − u + v ) ! . (3.35)Next, we turn to factor (3.34). Due to Lemma 3.2, we can assume that the integer p waschosen so that the coefficients of all monomials (3.28) in factor (3.34) coincide with thecoefficients of all monomials (3.28) in z p ( e z − p n Y i =1 m Y j =1 r ( e − z − u i + v j − h ( N + c ) ) g ( e − z − u i + v j ) . Moreover, by (3.12), this is equal to z p ( e z − p ψ nm n Y i =1 m Y j =1 g ( e z + u i − v j + h ( N + c ) ) g ( e − z − u i + v j ) . (3.36)Finally, we conclude that the coefficients of all monomials (3.28) in (3.32) coincide withthe coefficients of the corresponding monomials in the product of (3.35) and (3.36).Consider the expression z p ( e z − p Y W ( Y ( T +13[ n ] ( u ) , z ) T +23[ m ] ( v ) , z ) , (3.37)which corresponds to the second summand in (3.31). By (2.15) it is equal to z p ( e z − p Y W ( T +13[ n ] ( u | z ) T ∗ n ] ( u | z + hc/ − T +23[ m ] ( v ) , z ) . (3.38)Since T ∗ ( u ) = , by combining relation (2.11) and the first crossing symmetry relationin (2.17) we obtain T ∗ n ] ( u | z + hc/ − T +23[ m ] ( v ) = (cid:0) D n ] R nm ( e z + u − v + h ( N + c ) )( D n ] ) − (cid:1) · RL (cid:16) T +23[ m ] ( v ) R nm ( e z + u − v ) − (cid:17) . (3.39)Introduce the functions g ( u, v, z ) = ψ nm n Y i =1 m Y j =1 g ( e z + u i − v j + h ( N + c ) ) , g ( u, v, z ) = ψ nm n Y i =1 m Y j =1 g ( e − z − u i + v j ) . By (1.18) we have R nm ( e z + u − v + h ( N + c ) ) = g ( u, v, z ) R +12 nm ( e z + u − v + h ( N + c ) ) . (3.40)Furthermore, by combining (1.18) and unitarity property (1.19) we find R nm ( e z + u − v ) − = R nm ( e − z − u + v ) = g ( u, v, z ) R +21 nm ( e − z − u + v ) . (3.41)Using (3.40) and (3.41) we rewrite the right hand side of (3.39) as g ( u, v, z ) g ( u, v, z ) (cid:0) D n ] R +12 nm ( e z + u − v + h ( N + c ) )( D n ] ) − (cid:1) · RL (cid:16) T +23[ m ] ( v ) R +21 nm ( e − z − u + v ) (cid:17) . (3.42)Next, we employ (3.42) and then (0.3) to express (3.38) as (cid:0) D n ] R +12 nm ( e z + u − v + h ( N + c ) )( D n ] ) − (cid:1) · RL (cid:16) Y W ( T +13[ n ] ( u | z ) T +23[ m ] ( v ) , z ) R +21 nm ( e − z − u + v ) (cid:17) × z p ( e z − p g ( u, v, z ) g ( u, v, z ) (cid:0) D n ] R +12 nm ( e z + u − v + h ( N + c ) )( D n ] ) − (cid:1) · RL (cid:16) L [ n + m ] ( x, y ) (cid:12)(cid:12)(cid:12) x i = z e z ui y j = z e vj R +21 nm ( e − z − u + v ) (cid:17) × z p ( e z − p g ( u, v, z ) g ( u, v, z ) . Note that z p ( e z − p g ( u, v, z ) g ( u, v, z ) is equal to (3.36) and that L [ n + m ] ( x, y ) (cid:12)(cid:12)(cid:12) x i = z e z ui y j = z e vj = L [ n + m ] ( x, y ) (cid:12)(cid:12)(cid:12) x i = z e ui y j = z e vj ! (cid:12)(cid:12)(cid:12)(cid:12) z = z e z . Therefore, the product of (3.35) and (3.36) is equal to (3.37), so we conclude that thecoefficients of all monomials (3.28) in (3.32) and in (3.37) coincide, as required. (cid:3) Lemma 3.8. The map Y W ( z ) satisfies b S -locality (2.31) , i.e. for any u, v ∈ V c ( gl N ) and k ∈ Z > there exists p ∈ Z > such that for all w ∈ W ( z − z ) p Y W ( z ) (cid:0) ⊗ Y W ( z ) (cid:1) ι z ,z (cid:0) b S ( z /z )( u ⊗ v ) ⊗ w (cid:1) − ( z − z ) p Y W ( z ) (cid:0) ⊗ Y W ( z ) (cid:1) ( v ⊗ u ⊗ w ) ∈ h k W [[ z ± , z ± ]] . (3.43) Proof. Let n, m, a , . . . , a n , b , . . . , b m , k > u = ( u , . . . , u n ) and v = ( v , . . . , v m ) the families of variables. We will apply Y W ( z ) (cid:0) ⊗ Y W ( z ) (cid:1) ι z ,z b S ( z /z ),which corresponds to the first summand in (3.43), to T +13[ n ] ( u ) T +24[ m ] ( v )( ⊗ ) (3.44)and then consider the coefficients of all monomials (3.28) in the resulting expression. Notethat the superscripts 1 , , , z }| { (End C N ) ⊗ n ⊗ z }| { (End C N ) ⊗ m ⊗ z }| { V c ( gl N ) ⊗ z }| { V c ( gl N ) . Applying the map b S ( z /z ), given by (2.22), to (3.44) and then using (1.15) we get b G ( z /z , u, v ) (cid:0) D m ] R +12 nm ( z e u − v − h ( N + c ) /z ) − ( D m ] ) − (cid:1) (3.45) · LR (cid:16) R +12 nm ( z e u − v /z ) T +13[ n ] ( u ) R +12 nm ( z e u − v + hc /z ) − T +24[ m ] ( v ) R +12 nm ( z e u − v /z )( ⊗ ) (cid:17) , where the function b G ( x, u, v ) is given by Lemma 3.4. As we only consider the coefficientsof monomials (3.28), it is sufficient to carry out the calculations modulo U , where U = n X i =1 u a i i V + m X j =1 v b j j V + h k V and V = (End C N ) ⊗ ( n + m ) ⊗ V c ( gl N ) ⊗ (( z ))(( z ))[[ u , . . . , u n , v , . . . , v m , h ]] . By Lemma 3.4, there exists an integer p > z − z ) p and (3.45) with respect to the map ι = ι z ,z ,u ,...,u n ,v ,...,v m coincides with( z − z ) p ι b H ( z /z , u, v ) (cid:0) D m ] R +12 nm ( z e u − v − h ( N + c ) /z ) − ( D m ] ) − (cid:1) (3.46) · LR (cid:16) R +12 nm ( z e u − v /z ) T +13[ n ] ( u ) R +12 nm ( z e u − v + hc /z ) − T +24[ m ] ( v ) R +12 nm ( z e u − v /z )( ⊗ ) (cid:17) modulo U , where the function b H ( x, u, v ) is given by Lemma 3.4. Note that there areonly finitely many monomials (3.28). Therefore, due to Lemma 3.2, we can assume that p = 4 p for some integer p > ι (cid:0) ( z − z ) p R nm ( z e u − v − h ( N + c ) /z ) − (cid:1) , ι (cid:0) ( z − z ) p R nm ( z e u − v /z ) (cid:1) ,ι (cid:0) ( z − z ) p R nm ( z e − u + v − hc /z ) (cid:1) , ι (cid:0) ( z − z ) p R nm ( z e − u + v /z ) − (cid:1) elong to (End C N ) ⊗ ( n + m ) (( z , z )). Due to the definition of the function b H ( x, u, v ), seein particular (3.16), we conclude by (1.15) and (3.2) that the expression in (3.46) equals( z − z ) p ι (cid:0) D m ] R nm ( z e u − v − h ( N + c ) /z ) − ( D m ] ) − (cid:1) · LR (cid:16) R nm ( z e u − v /z ) (3.47) × T +13[ n ] ( u ) R nm ( z e − u + v − hc /z ) T +24[ m ] ( v ) R nm ( z e − u + v /z ) − ( ⊗ ) (cid:17) mod U. Next, we apply Y W ( z )(1 ⊗ Y W ( z )) to (3.47), thus getting( z − z ) p (cid:0) D m ] ι (cid:0) R nm ( z e u − v − h ( N + c ) /z ) − (cid:1) ( D m ] ) − (cid:1) · LR (cid:16) ι (cid:0) R nm ( z e u − v /z ) (cid:1) (3.48) × L n ] ( x ) (cid:12)(cid:12) x i = z e ui ι (cid:0) R nm ( z e − u + v − hc /z ) (cid:1) L m ] ( y ) (cid:12)(cid:12) y j = z e vj ι (cid:0) R nm ( z e − u + v /z ) − (cid:1) (cid:17) modulo U , where x = ( x , . . . , x n ) and y = ( y , . . . , y m ) are the families of variables and U = n X i =1 u a i i V + m X j =1 v b j j V + h k V for V = (End C N ) ⊗ ( n + m ) ⊗ Hom( W, W [[ z ± , z ± ]])[[ u , . . . , u n , v , . . . , v m ]] . By employing quantum current commutation relation (1.34) we rewrite (3.48) as (cid:0) D m ] ι (cid:0) ( z − z ) p R nm ( z e u − v − h ( N + c ) /z ) − (cid:1) ( D m ] ) − (cid:1) · LR (cid:16) ι (cid:0) ( z − z ) p R nm ( z e u − v /z ) (cid:1) ι (cid:0) ( z − z ) p R nm ( z e u − v /z ) − (cid:1) (3.49) × L m ] ( y ) (cid:12)(cid:12) y j = z e vj ι (cid:0) ( z − z ) p R nm ( z e u − v − hc /z ) (cid:1) L n ] ( x ) (cid:12)(cid:12) x i = z e ui (cid:17) mod U . Observe that all products in (3.49) are well-defined modulo U due to our choice of theinteger p = 4 p and Remark 1.5. Canceling the R -matrices R nm ( z e u − v /z ) ± and thenusing the following consequence of the second crossing symmetry relation in (1.13) whichis verified by arguing as in Remark 2.4, (cid:0) D m ] R nm ( z e u − v − h ( N + c ) /z ) − ( D m ] ) − (cid:1) · LR R nm ( z e u − v − hc /z ) , the expression in (3.49) simplifies to( z − z ) p L m ] ( y ) (cid:12)(cid:12) y j = z e vj L n ] ( x ) (cid:12)(cid:12) x i = z e ui mod U . (3.50)Finally, consider the expression which corresponds to the second summand in (3.43),i.e. which is obtained by applying ( z − z ) p Y W ( z )(1 ⊗ Y W ( z )) to T +14[ n ] ( u ) T +23[ m ] ( v )( ⊗ ).Clearly, all its coefficients with respect to monomials (3.28) coincide with the correspond-ing coefficients in (3.50), so the b S -locality follows. (cid:3) Establishing the restricted U h ( b gl N ) -module structure. Let ( W, Y W ) be a φ -coordinated V c ( gl N )-module for some c ∈ C , where φ ( z , z ) = z e z . In this subsection,which consists of two lemmas, we finish the proof of the Main Theorem in the gl N case. Lemma 3.9. Formula (0.4) defines a unique structure of restricted U h ( b gl N ) -module oflevel c on W . Proof. The uniqueness is clear as (0.4) determines the action of all generators of U h ( b gl N )on W . We now use the Jacobi-type identity given in Proposition 2.10 to check that (0.4)satisfies defining relation (1.27) for the algebra U h ( b gl N ) at the level c . Let n > p > ι z ,z ( z − z ) p R ( z /z ) − T +23 (0) R ( z e − hc /z ) T +14 (0)( ⊗ ) and (3.51) ι z ,z ( z − z ) p R ( z /z ) − T +23 (0) R ( z e − hc /z ) T +14 (0)( ⊗ ) , (3.52) hose coefficients belong to (End C N ) ⊗ ⊗ V c ( gl N ) ⊗ , coincide modulo h n . Note thatthe embedding map ι z ,z in (3.52) can be omitted as both R -matrices are Taylor seriesin z /z , i.e. they consist of nonnegative powers of z /z . Furthermore, we can assumethat the integer p is chosen so that expression (3.51) modulo h n is a polynomial in thevariables z ± , z ± . Hence the embedding map ι z ,z can be also omitted when regarding(3.51) modulo h n . Applying first term (2.32) of the Jacobi identity on (3.52) we get( z z ) − δ (cid:18) z − z z z (cid:19) Y W ( z )(1 ⊗ Y W ( z )) × ( z − z ) p R ( z /z ) − T +23 (0) R ( z e − hc /z ) T +14 (0)( ⊗ ) . Using (0.4) we rewrite this as( z z ) − δ (cid:18) z − z z z (cid:19) ( z − z ) p R ( z /z ) − L ( z ) R ( z e − hc /z ) L ( z ) . Due to the well-known δ -function identity, xδ ( x ) = δ ( x ) , (3.53)by multiplying by ( z z ) − p and then taking the residue Res z z we obtain R ( z /z ) − L ( z ) R ( z e − hc /z ) L ( z ) . (3.54)We now turn to second term (2.33) of the Jacobi identity. Choose r > A ( z , z ) := ι z ,z ( z − z ) r R ( z e hc /z ) − , B ( z , z ) := ι z ,z ( z − z ) r ψ R ( z e − hc /z ) ,C ( z , z ) := ι z ,z ( z − z ) r R ( z /z ) , D ( z , z ) := ι z ,z ( z − z ) r ψ − R ( z /z ) − belong to (End C N ) ⊗ (( z , z )) modulo h n . As with (3.52), observe that the embeddingmap ι z ,z can be omitted in the definitions of B ( z , z ) and D ( z , z ) above. By combining(1.18) and unitarity property (1.19) we find A ( z , z ) (cid:12)(cid:12) mod h n z = z e z = B ( z , z ) (cid:12)(cid:12) mod h n z = z e z and C ( z , z ) (cid:12)(cid:12) mod h n z = z e z = D ( z , z ) (cid:12)(cid:12) mod h n z = z e z . Therefore, by the implication in (3.11) we conclude that A ( z , z ) = B ( z , z ) mod h n and C ( z , z ) = D ( z , z ) mod h n . (3.55)Consider (3.51) modulo h n . Applying second term (2.33) of the Jacobi identity we get − ( z z ) − δ (cid:18) z − z − z z (cid:19) Y W ( z )(1 ⊗ Y W ( z )) ι z ,z (cid:0) b S ( z /z ) × ( z − z ) p R ( z /z ) − T +24 (0) R ( z e − hc /z ) T +13 (0)( ⊗ ) (cid:1) mod h n . (3.56)By using explicit formula (2.23) for the map b S ( x ) we rewrite (3.56) as − ( z z ) − δ (cid:18) z − z − z z (cid:19) Y W ( z )(1 ⊗ Y W ( z )) × ( z − z ) p T +13 (0) ι z ,z (cid:0) R ( z e hc /z ) − (cid:1) T +24 (0) ι z ,z ( R ( z /z )) ( ⊗ ) mod h n . Next, the application of (0.4) gives us − ( z z ) − δ (cid:18) z − z − z z (cid:19) ( z − z ) p L ( z ) × ι z ,z (cid:0) R ( z e hc /z ) − (cid:1) L ( z ) ι z ,z ( R ( z /z )) mod h n . (3.57) Note that, in contrast with (2.32) and (2.34), the vectors u and v in (2.33) are swapped, so that in(3.56) we have T +24 (0) and T +13 (0) instead of T +23 (0) and T +14 (0). ote that (3.53) implies δ (cid:18) z − z − z z (cid:19) = ( z − z ) r ( − z z ) r δ (cid:18) z − z − z z (cid:19) , so that we can use both equalities in (3.55) to rewrite (3.57) as − ( z z ) − δ (cid:18) z − z − z z (cid:19) ( z − z ) p L ( z ) R ( z e − hc /z ) L ( z ) R ( z /z ) − mod h n , where the embedding maps ι z ,z are omitted as both R -matrices consist of nonnegativepowers of z /z . Finally, multiplying by ( z z ) − p and taking the residue Res z z we get − L ( z ) R ( z e − hc /z ) L ( z ) R ( z /z ) − mod h n . (3.58)By applying third term (2.34) of the Jacobi identity to (3.51) we get z − δ (cid:18) z (1 + z ) z (cid:19) Y W ( z ) ( Y (log(1 + z )) ⊗ × ι z ,z ( z − z ) p R ( z /z ) − T +23 (0) R ( z e − hc /z ) T +14 (0)( ⊗ ) mod h n . (3.59)As before, by (1.18) and unitarity property (1.19) there exists s > ι z ,z ( z − z ) s R ( z e − hc /z ) = ι z ,z ( z − z ) s ψ − R ( z e hc /z ) − mod h n . (3.60)Using the δ -function identities (cid:18) z z (cid:19) l δ (cid:18) z (1 + z ) z (cid:19) = (1 + z ) l δ (cid:18) z (1 + z ) z (cid:19) , (3.61)which follow directly from (3.53), one can easily derive δ (cid:18) z (1 + z ) z (cid:19) = ( z − z ) k ( z z ) k δ (cid:18) z (1 + z ) z (cid:19) , in particular for k = p, s . Therefore, we can employ (3.60) to rewrite (3.59) as ψ − z − δ (cid:18) z (1 + z ) z (cid:19) ( z z ) p Y W ( z ) ( Y (log(1 + z )) ⊗ × ι z ,z (cid:0) R ( z /z ) − (cid:1) T +23 (0) ι z ,z (cid:0) R ( z e hc /z ) − (cid:1) T +14 (0)( ⊗ ) mod h n . Next, using definition (2.15) of the vertex operator map and (3.61) we get ψ − z − δ (cid:18) z (1 + z ) z (cid:19) ( z z ) p Y W ( z ) ι z ,z (cid:0) R ( z /z ) − (cid:1) × T +23 (log(1 + z )) T ∗ (log(1 + z ) + hc/ − R ((1 + z ) e hc ) − T +13 (0) mod h n . Finally, we use relation (2.11) to swap the operators T ∗ and T +13 , and then we employthe identity T ∗ ( x ) = , thus getting ψ − z − δ (cid:18) z (1 + z ) z (cid:19) ( z z ) p Y W ( z ) ι z ,z (cid:0) R ( z /z ) − (cid:1) × T +23 (log(1 + z )) T +13 (0) ι z ,z (cid:0) R ( z /z ) − (cid:1) mod h n . (3.62)It is clear that the application of the module map Y W ( z ) in (3.62) will not produceany negative powers of the variable z . Therefore, multiplying (3.62) by ( z z ) − p and thentaking the residue Res z z we obtain 0 mod h n . Hence combining the Jacobi-type identityfrom Proposition 2.10 with (3.54) and (3.58) we obtain the equality L ( z ) R ( z e − hc /z ) L ( z ) R ( z /z ) − − R ( z /z ) − L ( z ) R ( z e − hc /z ) L ( z ) = 0 mod h n or operators on W . As the integer n was arbitrary, we conclude that the given equalityholds for all n . Hence we proved that (0.4) satisfies quantum current commutation relation(1.27), so that it defines the structure of U h ( b gl N )-module of level c on W , as required. Inthe end, in order to finish the proof, it remains to observe that W is a topologically free C [[ h ]]-module and, furthermore, restricted U h ( b gl N )-module by Definition 2.7. (cid:3) The next lemma completes the proof of the Main Theorem for g N = gl N . Lemma 3.10. A topologically free C [[ h ]] -submodule W of W is a φ -coordinated V c ( gl N ) -submodule of W if and only if it is an U h ( b gl N ) -submodule of W . Proof. Suppose that W is a φ -coordinated V c ( gl N )-submodule of W . Then L ( z ) w = Y W ( T + (0) , z ) w ∈ End C N ⊗ W (( z ))[[ h ]] for any w ∈ W , so W is clearly an U h ( b gl N )-submodule of W .Conversely, suppose that W is a topologically free U h ( b gl N )-submodule of W . Clearly, W is a restricted U h ( b gl N )-module of level c , so by Proposition 1.4 we have L [ n ] ( x , . . . , x n ) w ∈ (cid:0) End C N (cid:1) ⊗ n ⊗ W (( x , . . . , x n ))[[ h ]] for all n > w ∈ W . Applying the substitutions x i = ze u i with i = 1 , . . . , n we get L [ n ] ( x ) (cid:12)(cid:12) x i = ze ui w = Y W ( T +[ n ] ( u ) , z ) w ∈ (cid:0) End C N (cid:1) ⊗ n ⊗ W (( z ))[[ u , . . . , u n , h ]] . By [9, Sect. 3.4], see also [26, Prop. 2.4], the coefficients of all matrix entries of T +[ n ] ( u ), n > 1, and span an h -adically dense C [[ h ]]-submodule of V c ( gl N ), so we conclude that W is a φ -coordinated V c ( gl N )-submodule of W , as required. (cid:3) Proof of the Main Theorem in the sl N case. For any integer n = 1 , . . . , N set u [ n ] = ( u, u − h, . . . , u − ( n − h ) and x [ n ] = ( x, xe − h , . . . , xe − ( n − h ) . Let P ( n ) : x ⊗ . . . ⊗ x n x n ⊗ . . . ⊗ x be the permutation operator on ( C N ) ⊗ n . Write L [ n ] ( x [ n ] ) = L [ n ] ( x , . . . , x n ) (cid:12)(cid:12) x = x,...,x n = xe − ( n − h ,~ L [ n ] ( x [ n ] ) = P ( n ) L [ n ] ( x n , . . . , x ) (cid:12)(cid:12) x = x,...,x n = xe − ( n − h P ( n ) . We first list some useful properties of the anti-symmetrizer A ( n ) defined by (1.35). Lemma 3.11. For any n = 1 , . . . , N we have A ( n ) L [ n ] ( x [ n ] ) = ~ L [ n ] ( x [ n ] ) A ( n ) , (3.63) A ( n ) D . . . D n = D . . . D n A ( n ) , (3.64) A ( N ) R N ( y/x [ N ] ) = ~R N ( y/x [ N ] ) A ( N ) = A ( N ) e − ( N − h/ x − e ( N − h yx − y , (3.65) where the arrow in ~R N ( y/x [ N ] ) indicates the reversed order of factors. The coefficientsin (3.65) belong to End C N ⊗ (End C N ) ⊗ N [[ h ]] and the anti-symmetrizer A ( N ) is appliedon the tensor factors , . . . , N + 1 . Proof. Equality (3.63) is verified by using Yang–Baxter equation (1.3), generalized quan-tum current commutation relation (1.34) and the following case of the fusion procedurefor the two-parameter R -matrix R ( x, y ) = ( xe − h/ − ye h/ ) R ( x/y ) going back to [4], −→ Y i =1 ,...,n − −→ Y j = i +1 ,...,n R ij ( xe − ( i − h , xe − ( j − h ) = n ! x n ( n − Y i Formula (0.3) , together with Y W ( , z ) = 1 W , defines a unique structureof φ -coordinated V c ( sl N ) -module on W , where φ ( z , z ) = z e z . Proof. In order to prove the lemma, it is sufficient to verify that (0.3), together with Y W ( , z ) = 1 W , defines a C [[ h ]]-module map V c ( sl N ) ⊗ W → W (( z ))[[ h ]]. Indeed, allother properties of the aforementioned map are recovered by arguing as in the g N = gl N case; see Subsection 3.2. Therefore, we have to show that the map v Y W ( v, z ) preservesthe ideal of relations (2.1) and (2.4). However, it is sufficient to consider (2.4) as relations(2.1) are already taken care of in the proof of Lemma 3.5.Let n and m be nonnegative integers. Introduce the families of variables v = ( v , . . . , v n )and w = ( w , . . . , w m ). Consider the image of the expression T +13[ n ] ( v )qdet T + ( u ) T +23[ m ] ( w ) ∈ (End C N ) ⊗ ( n + m ) ⊗ V c ( sl N )[[ v , . . . , v n , u, w , . . . w m ]](3.67)with respect to Y W ( z ). Introduce the tensor product z }| { (End C N ) ⊗ n ⊗ z }| { (End C N ) ⊗ N ⊗ z }| { (End C N ) ⊗ m ⊗ z }| { V c ( sl N ) (3.68)and write A ( N )2 = 1 ⊗ n ⊗ A ( N ) ⊗ ⊗ m and D N ] = 1 ⊗ n ⊗ D [ N ] ⊗ ⊗ m = D n +1 . . . D n + N . By the definition of quantum determinant given by (2.2), using the labels in (3.68) toindicate the corresponding tensor factors, (3.67) can be expressed astr n +1 ,...,n + N A ( N )2 T +14[ n ] ( v ) T +24[ N ] ( u [ N ] ) T +34[ m ] ( w ) D N ] . (3.69)By (0.3), the image of (3.69) with respect to Y W ( z ) equalstr n +1 ,...,n + N A ( N )2 L [ n + N + m ] ( x, x [ N ] , y ) (cid:12)(cid:12)(cid:12) x = ze v , ..., x n = ze vn , x = ze u y = ze w , ..., y n = ze wm D N ] , (3.70)where x = ( x , . . . , x n ) and y = ( y , . . . , y m ) . Using generalized quantum current commu-tation relation (1.34) we transform A ( N )2 L [ n + N + m ] ( x, x [ N ] , y ) and bring it to the form A ( N )2 L n ] ( x ) R nN ( x [ N ] e − hc /x ) R nm ( ye − hc /x ) L N ] ( x [ N ] ) × R Nm ( ye − hc /x [ N ] ) L m ] ( y ) R Nm ( y/x [ N ] ) − R nm ( y/x ) − R nN ( x [ N ] /x ) − . (3.71)By employing (1.11) and (3.65) one can verify the following identities: A ( N )2 R nN ( x [ N ] e − hc /x ) = e − n ( N − h/ A ( N )2 , A ( N )2 R nN ( x [ N ] /x ) − = e n ( N − h/ A ( N )2 ,A ( N )2 R Nm ( ye − hc /x [ N ] ) = e − m ( N − h/ A ( N )2 , A ( N )2 R Nm ( y/x [ N ] ) − = e m ( N − h/ A ( N )2 . As the anti-symmetrizer A ( N )2 commutes with the terms R nm ( ye − hc /x ), R nm ( y/x ) − , L n ] ( x ) and L m ] ( y ), by combining the above identities and (3.63), we rewrite (3.71) as L n ] ( x ) R nm ( ye − hc /x ) A ( N )2 L N ] ( x [ N ] ) L m ] ( y ) R nm ( y/x ) − . (3.72)Note that the expression in (3.70) is obtained from (3.72) by applying the substitutions x = ze v , . . . , x n = ze v n , x = ze u , y = ze w , . . . , y n = ze w m , hen multiplying by D N ] from the right and, finally, taking the trace tr n +1 ,...,n + N . However,as D N ] commutes with the terms L m ] ( y ) and R nm ( y/x ) − , it is clear that applying theaforementioned transformations to (3.72) and using definition of quantum determinant(1.36) results in L n ] ( x ) R nm ( ye − hc /x )qdet L ( x ) L m ] ( y ) R nm ( y/x ) − (cid:12)(cid:12)(cid:12) x = ze v , ..., x n = ze vn , x = ze u y = ze w , ..., y n = ze wm , (3.73)where, due to application of the trace, the tensor factors in (3.73) are now labeled inaccordance with (3.67). As qdet L ( x ) = 1 in U h ( b sl N ) we conclude by quantum currentcommutation relation (1.34) that (3.73) is equal to L [ n + m ] ( x, y ) (cid:12)(cid:12)(cid:12) x = ze v , ..., x n = ze vn y = ze w , ..., y n = ze wm = Y W ( T +13[ n ] ( v ) T +23[ m ] ( w ) , z ) . Therefore, the images of (3.67) and T +13[ n ] ( v ) T +23[ m ] ( w ) with respect to Y W ( z ) coincide, sowe conclude that the C [[ h ]]-module map V c ( sl N ) ⊗ W → W (( z ))[[ h ]] is well-defined by(0.3), as required. (cid:3) Let ( W, Y W ) be a φ -coordinated V c ( sl N )-module for some c ∈ C , where φ ( z , z ) = z e z .In order to prove that (0.4) defines a unique structure of restricted U h ( b sl N )-module oflevel c on W , we need the following identity. Lemma 3.13. For any positive integer n the identity Y W (cid:0) ( T +1 (( n − h ) T +2 (( n − h ) . . . T + n (0) , ze − ( n − h (cid:1) = L [ n ] ( x , x , . . . , x n ) (cid:12)(cid:12) x = z,x = ze − h ,...,x n = ze − ( n − h (3.74) holds for operators on W , where the action of L [ n ] ( x , . . . , x n ) on W is given by formula (1.33) with L ( x ) = Y W ( T + (0) , x ) . Proof. We derive (3.74) using the weak associativity property. Let k be a positive integer.By (2.29) and (2.30) there exists an integer p > z − z ) p Y W ( T +1 (0) , z ) Y W ( T +2 (0) , z ) = ( z − z ) p L ( z ) L ( z )belongs to (End C N ) ⊗ ⊗ Hom( W, W (( z , z ))) modulo h k and such that(( z − z ) p L ( z ) L ( z )) (cid:12)(cid:12) mod h k z = z e z and z p ( e z − p Y W ( Y ( T +1 (0) , z ) T +2 (0) , z )(3.75)coincide modulo h k . Using relation (2.11) and then the first crossing symmetry propertyin (2.17) we express the second term in (3.75) as z p ( e z − p ( D R ( e z + h ( c + N ) ) D − ) · RL (cid:0) Y W ( T +1 ( z ) T +2 (0) , z ) R ( e z ) − (cid:1) . (3.76)The first crossing symmetry property in (2.17) and unitarity (1.19) imply the identities R ( e − z − hc ) · RL ( D R ( e z + h ( c + N ) ) D − ) = 1 and R ( e z ) − R ( e − z ) − = 1 , which enable us to move the R -matrices appearing in (3.76) from the second term in(3.75) to the first term in (3.75). Hence we find that (cid:16) R ( e − z − hc ) · RL (( z − z ) p L ( z ) L ( z )) (cid:12)(cid:12) mod h k z = z e z (cid:17) R ( e − z ) − (3.77)and z p ( e z − p Y W ( T +1 ( z ) T +2 (0) , z ) (3.78)coincide modulo h k . Without loss of generality we can assume that the integer p is suffi-ciently large, so that we conclude by Lemma 3.2 that (3.77) is equal to (cid:0) ( z − z ) p L ( z ) R ( z e − hc /z ) L ( z ) R ( z /z ) − (cid:1) (cid:12)(cid:12) mod h k z = z e z . (3.79) y employing (1.33) for n = 2 and the relation L [2] ( z , z ) ∈ (End C N ) ⊗ ⊗ Hom( W, W (( z , z ))[[ h ]]) , which is verified by arguing as in the proof of Proposition 1.4, we rewrite (3.79) as (cid:0) ( z − z ) p L [2] ( z , z ) (cid:1) (cid:12)(cid:12) mod h k z = z e z = z p ( e z − p (cid:0) L [2] ( z , z ) (cid:1) (cid:12)(cid:12) mod h k z = z e z . (3.80)Thus we proved that (3.78) and (3.80) coincide modulo h k . Hence, multiplying (3.78) and(3.80) by z − p ( e z − − p we find that L [2] ( z , z ) (cid:12)(cid:12) z = z e z and Y W ( T +1 ( z ) T +2 (0) , z )coincide modulo h k . Moreover, by setting z = h and z = ze − h we conclude that L [2] ( z , z ) (cid:12)(cid:12) z = z, z = ze − h and Y W ( T +1 ( h ) T +2 (0) , ze − h )coincide modulo h k . As the integer k > n = 2. The general case is proved by induction on n . (cid:3) The next two lemmas complete the proof of the Main Theorem for g N = sl N . Thesecond lemma follows by the same arguments as for Lemma 3.10, so we omit its proof. Lemma 3.14. Formula (0.4) defines a unique structure of restricted U h ( b sl N ) -module oflevel c on W . Proof. Due to the proof of Lemma 3.9, it is sufficient to verify the equality qdet L ( z ) = 1on W , where the action of L ( z ) on W is given by (0.4). By (1.36) and (3.74), the actionof quantum determinant of L ( z ) on W is given bytr ,...,N A ( N ) Y W (cid:0) ( T +1 (( N − h ) T +2 (( N − h ) . . . T + N (0) , ze − ( N − h (cid:1) D . . . D N = Y W (cid:0) tr ,...,N A ( N ) ( T +1 (( N − h ) T +2 (( N − h ) . . . T + N (0) D . . . D N , ze − ( N − h (cid:1) . By applying (2.2) with u = ( N − h and (2.4) the given expression takes the form Y W (cid:0) qdet T + (( N − h ) , ze − ( N − h (cid:1) = Y W (cid:0) , ze − ( N − h (cid:1) . Finally, Definition 2.7 implies Y W (cid:0) , ze − ( N − h (cid:1) = 1, which completes the proof. (cid:3) Lemma 3.15. A topologically free C [[ h ]] -submodule W of W is a φ -coordinated V c ( sl N ) -submodule of W if and only if W is an U h ( b sl N ) -submodule of W . Image of the center of the quantum affine vertex algebra In this section, we briefly discuss a connection between families of central elementsfor the quantum affine vertex algebra and the quantum affine algebra established by the φ -coordinated module map from the Main Theorem.4.1. Noncritical level. Following [22], we define the center of the quantum vertex al-gebra V c ( gl N ) at the level c ∈ C as the C [[ h ]]-submodule z ( V c ( gl N )) = (cid:8) v ∈ V c ( gl N ) : Y ( w, z ) v ∈ V c ( gl N )[[ z ]] for all w ∈ V c ( gl N ) (cid:9) . For more details on the notion of center of quantum vertex algebra see [5, Thm. 1.4]and [22, Sect. 3.2]. Observe that (0.3) implies the identity Y W (qdet T + (0) , z ) = qdet L ( z ) (4.1)on any restricted U h ( b gl N )-module W of level c ∈ C . By [26, Prop. 3.10] the coefficientsof the quantum determinant qdet T + ( u ), as given by (2.3), belong to the center of thequantum vertex algebra V c ( gl N ) for any c ∈ C . The next proposition, which is well-known, provides a quantum affine algebra counterpart of this fact; cf. [13]. We formulate he proposition and outline its proof in terms of Ding’s quantum current realization forcompleteness. Proposition 4.1. For any c ∈ C all coefficients d r of the quantum determinant qdet L ( z ) ,as given by (1.37) , belong to the center of the quantum affine algebra U h ( b gl N ) c . Proof. It is sufficient to prove the equality L ( y ) qdet L ( x ) = qdet L ( x ) L ( y ) (4.2)in End C N ⊗ U h ( b gl N ) c . By (1.36) the left hand side in (4.2) equalstr ,...,N L ( y ) A ( N ) L [ N ] ( x [ N ] ) D [ N ] , where D [ N ] = D . . . D N (4.3)and the coefficients of the expression under the trace belong to the tensor productEnd C N ⊗ (End C N ) ⊗ N ⊗ U h ( b gl N ) c . The copies of End C N in (4.3) are labeled by 0 , . . . , N .The matrix L ( y ) is applied on the tensor factor 0 while the remaining terms, A ( N ) , L [ N ] ( x [ N ] ) and D [ N ] are applied on the tensor factors 1 , . . . , N . By L ( y ) A ( N ) = A ( N ) L ( y )and generalized quantum current commutation relation (1.34) we rewrite (4.3) astr ,...,N A ( N ) (cid:16) A · RL (cid:0)(cid:0) B L [ N ] ( x [ N ] ) C L ( y ) (cid:1) E (cid:1)(cid:17) D [ N ] , (4.4)where A = D − N ] R N ( x [ N ] e − ( N + c ) h /y ) − D [ N ] , B = R N ( y/x [ N ] ) − ,C = R N ( ye − hc /x [ N ] ) , E = R N ( x [ N ] /y ) . Note that the element A is found via the second crossing symmetry property in (1.13); seealso Remark 2.4. Next, by using (1.11) and (3.65) one can verify the following equalities: A ( N ) Z = λ Z A ( N ) for Z = A, B, C, E and λ A = λ B = λ − C = λ − E = e ( N − h/ . (4.5)Using (3.63) and (4.5) we move the anti-symmetrizer in (4.4) to the right, thus gettingtr ,...,N ~ L [ N ] ( x [ N ] ) L ( y ) A ( N ) D [ N ] = tr ,...,N ~ L [ N ] ( x [ N ] ) A ( N ) D [ N ] L ( y ) . Finally, we use (3.63) to move the anti-symmetrizer A ( N ) to the left, thus getting theright hand side in (4.2), as required. (cid:3) Following [14, Sect. 3.3], we define the submodule of invariants of the vacuum module V c ( gl N ) as the C [[ h ]]-submodule z ( V c ( gl N )) = { v ∈ V c ( gl N ) : L ( z ) v ∈ V c ( gl N )[[ z ]] } . Recall Corollary 0.1. By setting W = V c ( gl N ) in (4.1) and then applying the resultingequality on 1 ∈ V c ( gl N ) one recovers the invariants of the vacuum module; cf. [13]. Corollary 4.2. For any c ∈ C all coefficients of the series ℓ N ( z ) := Y V c ( gl N ) (qdet T + (0) , z )1 = qdet L ( z )1 ∈ V c ( gl N )[[ z ]] . belong to the submodule of invariants z ( V c ( gl N )) . Proof. The Corollary follows by applying identity (4.2) on 1 ∈ V c ( gl N ). (cid:3) Critical level. Consider the quantum affine vertex algebra at the critical level V cri ( gl N ) = V − N ( gl N ). The following family of central elements for the quantum vertexalgebra V cri ( gl N ) was given by Molev and the author [26, Prop. 3.5]. Proposition 4.3. All coefficients of the series φ n ( u ) := tr ,...,n A ( n ) T +[ n ] ( u, u − h, . . . , u − ( n − h ) D . . . D n ∈ V cri ( gl N )[[ u ]] with n = 1 , . . . , N belong to the center of the quantum vertex algebra V cri ( gl N ) . ow consider the quantum affine algebra at the critical level U h ( b gl N ) cri = U h ( b gl N ) − N .The next theorem goes back to Frappat, Jing, Molev and Ragoucy [13, Thm. 3.2]. Al-though it is originally given in terms of the RLL realization of the quantum affine algebra,we formulate the theorem using Ding’s quantum current realization. The direct proof interms of Ding’s realization is carried out by arguing as in the proof of [27, Thm. 2.14]and using Lemma 3.11. Theorem 4.4. All coefficients of the series ℓ n ( z ) := tr ,...,n A ( n ) L [ n ] ( z , . . . , z n ) (cid:12)(cid:12) z = z,...,z n = ze − ( n − h D . . . D n ∈ U h ( b gl N ) cri [[ z ± ]] with n = 1 , . . . , N belong to the center of the algebra U h ( b gl N ) cri . Finally, let W be any restricted U h ( b gl N )-module of level − N . Then the identities Y W ( φ n (0) , z ) = ℓ n ( z ) for n = 1 , . . . , N (4.6)hold for operators on W , where the map Y W ( z ) is given by (0.3). Recall Corollary 0.1. Bysetting W = V cri ( gl N ) in (4.6) and then applying the resulting equality on 1 ∈ V cri ( gl N )one recovers the invariants of the vacuum module; see [13, Corollary 3.3]. Corollary 4.5. All coefficients of the series ℓ n ( z ) := Y V cri ( gl N ) ( φ n (0) , z )1 = ℓ n ( z )1 ∈ V cri ( gl N )[[ z ]] with n = 1 , . . . , N belong to the submodule of invariants z ( V cri ( gl N )) . Acknowledgement The author would like to thank Naihuan Jing and Mirko Primc for stimulating discus-sions. The research reported in this paper was finalized during the author’s visit to MaxPlanck Institute for Mathematics in Bonn. The author is grateful to the Institute forits hospitality and financial support. This work has been supported in part by CroatianScience Foundation under the project 8488. References [1] B. Bakalov, V. G. Kac, Field algebras , Int. Math. Res. Not. (2003), no. 3, 123–159;arXiv:math/0204282 [math.QA].[2] R. Borcherds Vertex algebras, Kac–Moody algebras, and the Monster , Proc. Natl. Acad. Sci. USA (1986) 3068–3071.[3] M. Butorac, N. Jing, S. Koˇzi´c, h -Adic quantum vertex algebras associated with rational R -matrix intypes B , C and D , Lett. Math. Phys. (2019), 2439–2471; arXiv:1904.03771 [math.QA].[4] I. V. Cherednik, A new interpretation of Gelfand–Tzetlin bases , Duke Math. J. (1987), 563–577.[5] A. De Sole, M. Gardini, V. G. Kac, On the structure of quantum vertex algebras , J. Math. Phys. (2020), 011701 (29pp); arXiv:1906.05051 [math.QA].[6] J. Ding, Spinor Representations of U q ( ˆ gl ( n )) and Quantum Boson-Fermion Correspondence , Comm.Math. Phys. (1999), 399–420; arXiv:q-alg/9510014.[7] J. Ding, I. B. Frenkel, Isomorphism of two realizations of quantum affine algebra U q ( b gl ( n )), Comm.Math. Phys. (1993), 277–300.[8] J. Ding, K. Iohara, Generalization of Drinfeld Quantum Affine Algebras , Lett. Math. Phys. (1997),181–193; arXiv:q-alg/9608002.[9] P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, III , Selecta Math. (N.S.) (1998), 233–269;arXiv:q-alg/9610030.[10] P. Etingof and D. Kazhdan, Quantization of Lie bialgebras, IV , Selecta Math. (N.S.) (2000),79–104; arXiv:math/9801043 [math.QA].[11] P. Etingof, D. Kazhdan, Quantization of Lie bialgebras, V , Selecta Math. (N.S.) (2000), 105–130;arXiv:math/9808121 [math.QA].[12] N. Yu. Reshetikhin, L. A. Takhtadzhyan and L. D. Faddeev, Quantization of Lie groups and Liealgebras , Algebra i Analiz (1989), no. 1, 178–206 (Russian); English transl., Leningrad Math. J. (1990), no. 1, 193–225. 13] L. Frappat, N. Jing, A. Molev and E. Ragoucy, Higher Sugawara operators for the quantum affinealgebras of type A , Comm. Math. Phys. (2016), 631–657; arXiv:1505.03667 [math.QA].[14] E. Frenkel, Langlands correspondence for loop groups , Cambridge Studies in Advanced Mathematics,103. Cambridge University Press, Cambridge, 2007.[15] E. Frenkel, D. Ben-Zvi, Vertex Algebras, Algebraic Curves , Mathematical Surveys and Monographs,vol. 88, Second ed., American Mathematical Society, Providence, RI, 2004.[16] E. Frenkel, N. Reshetikhin, Towards deformed chiral algebras , preprint arXiv:q-alg/9706023.[17] I. B. Frenkel, N. Jing, Vertex representations of quantum affine algebras , Proc. Natl. Acad. Sci.USA, (1988), 9373–9377.[18] I. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the Monster , Pure and AppliedMathematics, 134. Academic Press, Inc., Boston, MA, 1988.[19] I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations ,Comm. Math. Phys. (1992), 1–60.[20] I. B. Frenkel and Y.-C. Zhu, Vertex operator algebras associated to representations of affine andVirasoro algebras , Duke Math. J. (1992), 123–168.[21] M. Jimbo, A q -difference analogue of U(G) and the Yang–Baxter equation , Lett. Math. Phys. (1985) 63–69.[22] N. Jing, S. Koˇzi´c, A. Molev, F. Yang, Center of the quantum affine vertex algebra in type A , J.Algebra (2018), 138–186; arXiv:1603.00237 [math.QA].[23] V. G. Kac, Infinite-dimensional Lie algebras , 3rd ed., Cambridge University Press, Cambridge, 1990.[24] V. Kac, Vertex algebras for beginners , University Lecture Series, 10. American Mathematical Society,Providence, RI, 1997.[25] C. Kassel, Quantum Groups , Graduate texts in mathematics; vol. , Springer-Verlag, 1995.[26] S. Koˇzi´c, A. Molev, Center of the quantum affine vertex algebra associated with trigonometric R -matrix , J. Phys. A: Math. Theor. (2017) 325201 (21pp); arXiv:1611.06700 [math.QA].[27] S. Koˇzi´c, Quantum current algebras associated with rational R -matrix , Adv. Math. (2019),1072–1104; arXiv:1801.03543 [math.QA].[28] J. Lepowsky, H.-S. Li, Introduction to Vertex Operator Algebras and Their Representations , Progressin Math., Vol. 227, Birkhauser, Boston, 2004.[29] H.-S. Li, Axiomatic G -vertex algebras , Commun. Contemp. Math. (2003), 281–327;arXiv:math/0204308 [math.QA].[30] H.-S. Li, ~ -adic quantum vertex algebras and their modules , Comm. Math. Phys. (2010), 475–523; arXiv:0812.3156 [math.QA].[31] H.-S. Li, φ -Coordinated Quasi-Modules for Quantum Vertex Algebras , Comm. Math. Phys. (2011), 703–741; arXiv:0906.2710 [math.QA].[32] H.-S. Li, S. Tan, Q. Wang, Ding–Iohara algebras and quantum vertex algebras , J. Algebra (2018),182–214; arXiv:1706.03636 [math.QA].[33] B.-H. Lian, On the classification of simple vertex operator algebras , Comm. Math. Phys. (1994),307–357.[34] J. H. H. Perk, C. L. Schultz, New families of commuting transfer matrices in q -state vertex models ,Phys. Lett. A (1981), 407–410.[35] N. Yu. Reshetikhin, M. A. Semenov-Tian-Shansky, Central extensions of quantum current groups ,Lett. Math. Phys. (1990), 133–142.(1990), 133–142.